Bulletin T period CXXII de l quoteright Acad acute-e mie Serbe des Sciences et des A t-r sub s hyphen 2001 \ centerlineClasse des Sciences{ Bulletin math T acute-e . CXXII matiques de etl naturelles ’ Acad $ \acute{e} $ mie Serbe des Sciences et des A $ tSciences−r { maths } acute-e − 2001 matiques $ } comma N o 26 JOVAN KARAMATA \ centerline1 902 hyphen{ Classe 1 967 des Sciences math $ \acute{e} $ matiques et naturelles } M period TOMI C-acute \ centerlineopen parenthesis{ Sciences Presented math at the $ 4\ thacute Meeting{e} comma$ matiques held on May $ 25 , comma N$ 200 o26 1 closing} parenthesis Jovan Karamata comesBulletin from T .a CXXII Serbian de merchant l ’ Acad familye´ mie Serbe from desthe Sciences city of et des A t − rs − 2001 \ centerlineZemun comma{JOVAN which KARAMATA is almost threeClasse} centuries des Sciences old period math ..e´ Businessmatiques relationset naturelles of his family comma on the borders of the Austro hyphen HungarianSciences math and T-ue´ matiques rkish empires, N o and26 Serbia comma \ centerlinewere very well{1 known 902 − and1 widespread 967 } period ..JOVAN Jovan Karamata KARAMATA was born in Zagreb on February 1 comma 1 902 comma but he always considered1 902 - 1 967 Zemun as his native city period .... In ´ \ centerline1 9 14 he finished{M . thereTOMI the $ first\acute three{ classesC} $ of} highM school . TOMI periodC .. Because of the war and since Zemun was on( the Presented border at thatthe 4 t th ime Meeting comma ,.. held Karamata on May quoteright 25 , 200 1 )s father \ centerlinehas sent him{( commaJovan Presented Karamatat ogether at with comesthe his 4 brothers from th Meeting a and Serbian his sist , merchant held er comma on May family t o Switzerland 25 from , 200 the period 1 city ) } ..of In Zemun Lausanne comma, which .. is in almost .. 1 920 three comma centuries .. he finished old high . school Business oriented relations t owards of mathematics his family , Jovanand sciences Karamataon period the comes borders .... In from the of same thea Serbian year Austro he enrolled merchant- Hungarian at the family Faculty and of fromT − Engineeringu rkish the city empires of and Serbia , Zemunof Belgrade , whichwere University very i s almost well and comma known t hree aft and er c e widespreadfinished n t u r i e twos old years . . of Jovan\quad t echnical KaramataBusiness studies comma was relations born he in ofZagreb his family , moved to theon FacultyFebruary of Philosophy 1 , 1 902 hyphen, but he Department always considered of Mathematics Zemun comma as his where native city . In \noindenthe graduated1on 9 in 14 the 1 he 925 borders finished period .. there of Let the us the mention Austro first three that− heHungarian classes submitted of high andhis doctoral school $ T−u .the $ hyphen Because rkish empiresof the war and Serbia , weresis three very monthsand well since after known Zemun his graduation and was widespread on period the border .. Obviously . \quad at that commaJovan t ime .. Karamata during, Karamata the waslast year born ’ s offather in Zagreb has sent his studieshim he was , t involved ogether in with active his research brothers comma and and his he sist studied er , t the o Switzerland scientific . In Lausanne \noindentliterature period,on February in .. He 1 sp 920 ent 1 , the , 1 he years 902 finished 1 , 927 but hyphen high he school always 28 in Paris oriented considered comma t owards .. as Zemun a fellow mathematics as of the his Rocke native hyphen city . \ h f i l l In feller Foundationand sciences comma .and in 1928 In he the b ecame same Assistant year he for enrolled Mathematics at the at Faculty the of Engineering \noindentFaculty ofof1 Philosophy 9 Belgrade 14 he of finished UniversityBelgrade University there and , the period aft er first In finished 1 930 three he two was classes Assistantyears of of Pro t highechnical hyphen school studies . \ ,quad he Because of the warfessor and comma smoved i n c in e 1 Zemun 937 to the Associate was Faculty on Professor the of border Philosophy and comma at that- after Department the t ime end of , World\ ofquad Mathematics WarKaramata II comma , ’ in where s father he has sentgraduated him , t ogether in 1 925 with . Let his us brothers mention andthat hishe submitted sist er his, t doctoral o Switzerland the - . \quad In Lausannesis , \ threequad monthsin \quad after1 his 920 graduation , \quad he . finished Obviously high , during school the oriented last year t of owards his mathematics studies he was involved in active research , and he studied the scientific literature \noindent .and sciencesHe sp ent .the\ h years f i l l 1In 927 the - 28 same in Paris year , he enrolled as a fellow at of the the Rocke Faculty - feller of Engineering Foundation , and in 1928 he b ecame Assistant for Mathematics at the Faculty of \noindent Philosophyof Belgrade of UniversityBelgrade University and , aft . In er 1 930 finished he was two Assistant years Pro of -t fessorechnical , in 1studies , he moved to937 the Associate Faculty ofProfessor Philosophy and , after− Department the end of of World Mathematics War II , in , where he graduated in 1 925 . \quad Let us mention that he submitted his doctoral the −

\noindent sis three months after his graduation . \quad Obviously , \quad during the last year of his studies he was involved in active research , and he studied the scientific literature . \quad He sp ent the years 1 927 − 28 in Paris , \quad as a fellow of the Rocke − feller Foundation , and in 1928 he b ecame Assistant for Mathematics at the Faculty of Philosophy of Belgrade University . In 1 930 he was Assistant Pro − fessor , in 1 937 Associate Professor and , after the end of World War II , in 2 .... M period Tomi acute-c \noindent1 950 he b2 ecame\ h f iF-u l l M ll Professor . Tomi period $ \acute In 1 95{c 1} he$ was elected u-F ll Professor at the University of Geneva period \noindentIn 1933 he1 was 950 elected he b a Corresponding ecame $ F− Memberu $ ll of Professor the Yugoslav .Academy In 1 95 1 he was elected $ u−F $ ll Professor at the of Sciences and Arts open parenthesis Zagreb closing parenthesis period and in 1 936 a Corresponding Member of regia \noindentSocietas ScientiarumUniversity Bohemica of Geneva open parenthesis . Prague closing parenthesis period In 1 939 he b ecame a Correspond hyphen \ hspaceing member∗{\2f i comma l l } In and 1933 in 1948 he was a F-u elected ll Member a of Corresponding the Serbian Academy Member of Sciences of the YugoslavM . Tomi Academyc´ and Arts period1 950 he b ecame F − u ll Professor . In 1 95 1 he was elected u − F ll Professor at \noindentHe was onetheof of theSciences founders and of the Arts Mathematical ( Zagreb Institute ) . and of the in Serbian 1 936 a Corresponding Member of regia Academy ofUniversity Sciences and of Arts Geneva in 1 946. comma and the first Editor hyphen in hyphen Chief of the \noindentj ournal PublicationsSocietasIn 1933 de Scientiarum l quoteright he was elected Institut Bohemica aMath Corresponding acute-e ( Prague matique ) Member . open In parenthesis 1 of 939 the he Yugoslav deuxi b ecame e-grave Academy a me Correspond s acute-e rie− closing parenthesis periodof Sciences .. He had and Arts ( Zagreb ) . and in 1 936 a Corresponding Member of regia \noindenta great influenceSocietasing member on manyScientiarum , students and in Bohemica of 1948 mathematics a ( $ Prague F in−u Belgrade $ ) . ll In and Member 1 939Serbia he of b theecame Serbian a Correspond Academy - of Sciences andduring Arts theing . years member 1 946 hyphen , and 1 in 953 1948 period a F − u ll Member of the Serbian Academy of Sciences In 1954 heand became Arts member . of the Editorial Board and Director of the \ hspacej ournal∗{\ L quoterightf i l l }HeHe was Enseignement was one one of of Maththe the founders foundersacute-e matique of of the the from Mathematical Mathematical Geneva period Institute ..Institute His influence of the ofon Serbian the the Serbian form and contentAcademy of this of jSciences ournal was and significant Arts in period 1 946 , and the first Editor - in - Chief of the j \noindentIn 193 1 heournalAcademy married Publications Emilij of Sciences a Nikolaj de evi land ’ acute-cInstitut Arts sub in Math comma 1 946e´ matique who , and gave birththe ( deuxi tofirst theire` me Editortwo s sonse´ rie− )in .− HeChief o f the jand ournal a daughterhad Publications period a great .. His influence de wife l died ’ on Institut in many 1 959 period students Math $ of\acute mathematics{e} $ inmatique Belgrade ( deuxi and Serbia $ \grave{e} $ meAfter s $ a\ longacuteduring illness{e} the comma$ years r i Jovan e 1 ) 946 .Karamata\ -quad 1 953He died . had on August 14 comma 1 967 in Geneva period aHis great ashes influence rest in his native onIn 1954 many t own he studentsof became Zemun period member of mathematics of the Editorial in Belgrade Board and Director Serbia of the duringThe development thej ournal years of Lmathematical 1 ’ 946 Enseignement− 1 sciences953 . Math in Serbiae´ matique starts with from Mihailo Geneva . His influence on the Petrovi acute-cform open and parenthesis content of 1864 this hyphen j ournal 1943 was closing significant parenthesis . period .. Petrovi c-acute was a student of famous \ hspace ∗{\ f i l l } In 1954 he became member of the Editorial Board and Director of the french mathematiIn hyphen 193 1 he married Emilij a Nikolaj evi c´, who gave birth to their two sons cians : .. Chand period a daughter Hermite comma . His H period wife died Poincar in 1 acute-e 959 . sub comma E period Picard comma P period Painlev e-acute sub\noindent period .. Pj period ournalAfter Appell a L long ’ period Enseignement illness .. During , Jovan Math Karamata $ \acute died{ one} August$ matique 14 , 1 from 967 inGeneva Geneva . \ .quad His influence on the formhis studies andHis content in Paris ashes comma of rest this in .. at his jthe native ournal E-acute t ownwas cole Normale of significant Zemun Sup . e-acute . rieure comma .. his student fellows were : .. E periodThe Boreldevelopment period .. ofH period mathematical Lebesgue sciencescomma E in period Serbia Cartan starts period with .. Mihailo who marked Petrovi this epoch in mathe hyphenIn 193 1 hec´(1864 married− 1943) Emilij. Petrovi a Nikolajc´ was a evi student $ \ ofacute famous{c} french{ , } mathemati$ who gave - birth to their two sons and a daughter . \quad His wife died in 1 959 . matics periodcians .. Together : Ch with . Hermite a few colleagues , H . Poincar comma Petrovie´, E . Picardacute-c not , P only . Painlev broughte´ t. o SerbiaP . Appell scientific ideas. and During theories his from studies Europe in comma Paris but , also at the the spiritE´ cole of science Normale from Sup e´ rieure , his Afterthe end a long ofstudent the 1illness 9 th fellows and ,the Jovan bwere eginning Karamata: of E the . 20Borel died th century . on August Hperiod . Lebesgue .. 14 Young , 1 people , 967 E . in commaCartan Geneva . . who Histheir ashes studentsmarked r e commas t in this accepted h i epoch s n at new inive mathe ideas t own and - maticso theories f Zemun . comma . Together but their with t eachers a few attracted colleagues , Petrovi them alsoc´ asnot scientists only broughtcomma not t onlyo Serbia as mathematicians scientific ideas period and .. Jovan theories Karamata from Europecomma who , but also Theabandoned developmentthe t echnical spirit of sciences of mathematical science comma from .. wassciences the attracted end of in by the Serbia the 1 p 9 ersonality th starts and the and with b work eginning Mihailo of the 20 th Petroviof Mihailocentury $ Petrovi\acute acute-c .{c} Young sub( period people1864 .. But ,− .. their aft1943 er students World ) War , . acceptedI comma$ \quad .. new inPetrovi .. ideas 1 920 commaand $ \ theoriesacute .. Petrovi{c ,} c-acute$ was was a t student of famous french mathemati − ired but their t eachers attracted them also as scientists , not only as mathematicians . \noindentand the worldJovanc i a n t s o Karamata which : \quad he bCh elonged , who . Hermite with abandoned all his , Hsoul t . echnical comma Poincar was sciences vanishing $ \acute , period{ wase} .. attracted{ His, }$ byE . the Picard p , P . Painlev $ \acute{e} { . }$ \quad P . Appell . \quad During creative yearsersonality were 1900 and hyphen work 19 of 14 Mihailo comma and Petrovi his wellc´. hyphenBut known aft results er World belonged War t I o , the in 1 hispast studies period920 .. Research in , Paris Petrovi in the , \c´ areaquadwas of tat differentialired the and $ the equations\acute world{ commaE t} o which$ theory cole he of bNormale functions elonged Supand with all$ \ hisacute soul{e} $ r i e u r e , \quad his student fellows werein particular : \quad, was ofE functions vanishing . Borel defined . . \ byquad His entireH creative series . Lebesgue comma years was were , E replaced .1900 Cartan by- 19 new 14 . fields\ ,quad and hiswho well marked - known this epoch in mathe − maticsof investigation . results\quad andTogether belonged by new ideas t with o period the a fewpast In spite colleagues . of this Research comma , Petrovi inPetrovi the acute-carea $ of\ hadacute differential influence{c} $ on equations not only brought t o Serbia scientific, theory ideas of and functions theories and from in particular Europe , of but functions also the defined spirit by entire of science series , from was the end ofreplaced the 1 by 9 th new and fields the of b investigation eginning of and the by 20 new th ideas century . In spite . \quad of thisYoung , Petrovi people , their studentsc´ had influence , accepted on new ideas and theories , but their t eachers attracted them also as scientists , not only as mathematicians . \quad Jovan Karamata , who abandoned t echnical sciences , \quad was attracted by the p ersonality and work of Mihailo Petrovi $ \acute{c} { . }$ \quad But \quad aft er World War I , \quad in \quad 1 920 , \quad Petrovi $ \acute{c} $ was t i r e d and the world t o which he b elonged with all his soul , was vanishing . \quad His creative years were 1900 − 19 14 , and his well − known results belonged t o the past . \quad Research in the area of differential equations , theory of functions and in particular of functions defined by entire series , was replaced by new fields of investigation and by new ideas . In spite of this , Petrovi $ \acute{c} $ had influence on Jovan Karamata .... 3 \noindentthe researchJovan career Karamata of Jovan Karamata\ h f i l l comma3 esp ecially at the beginning period .. Kara hyphen mata accepted his advice that one should work in modern areas and seek \noindentsolutions bythe new research methods comma career and of in particular Jovan Karamata that one should , esp study ecially the con at hyphen the beginning . \quad Kara − matat emporary accepted lit erature his periodadvice .. Already that one as a should student comma work in he noticed modern the areas importance and ofseek solutionsthe Stieltj es by integral new methods comma and , he and used in this particular integral t o represent that one finite should and infi study hyphen the con − tnite emporary sums in a lit unique erature way period . \quad .. He masteredAlready this as tool a studentand used it , in he his noticed first the importance of theworks Stieltj andJovan also es later Karamata integral period .. Still , and comma he Karamata used this i s self integral hyphen taught t o represent in science comma finite and and he sought infi3 − nitefor the sums properthe in inspiration research a unique career in way the lit of . erature\ Jovanquad period KaramataHe mastered .. Perhaps , esp this he ecially did tool not at know and the the beginningused it in . his Kara first - workswords and of Abelmata also comma accepted later but he . his\ followedquad adviceStill them that : one .. , it Karamata shouldi s b etter work toi reads in self modern masters− taught than areas stu and in hyphen seek science solutions , and he sought fordents the period properby .. new Karamata inspiration methods kept , saying and in in that the particular his lit t eacher erature that of classical one . \ shouldquad analysisPerhaps study was the the he con did - not t emporary know the wordsfamous of booklit Abel of erature P , O-acute but . he lya Already followed and Szeg as o-dieresis thema student : 1\ subquad , he period noticedit F-r i s sub the b om etter importance this book to read he of learned mastersthe Stieltj what inthan es stu − dentsclassical . analysis\quadintegralKaramata was , called and he the kept used mathematical saying this integral that style : this .. o using represent t eacher minimal finite means of classical and infi - analysis nite sums was in a the t o acquireunique the greatest way possibility . He mastered of clear expression this tool and and natural used order it in inhis first works and also later \noindentproofs comma.famous without Still book , using Karamata of phrases P $ i such\ sacute self as .. -{O taughtquotedblright} $ lya in science and i .. i Szeg .. , eas and .. $ t he ..\ddotprove sought{o quotedblright} for1 the{ proper. ..} an ..F− quotedblrightr { om }$ obthis vou book sly inspiration he learned in what the lit in erature . Perhaps he did not know the words of Abel , but classicalperiod periodhe analysis followed period Undoubtedly themwas called : at it that i the s b t etterime mathematical comma to read around masters style 1 925 than comma : \quad stu there -using dents existed . minimal important Karamata means results tproved o acquire bykept ingenious the saying greatest and that beautiful his possibility t techniques eacher of comma classical of clear which analysis could expression serve was as the a and model natural order in proofs , without using phrases´ such as \quad ’ ’ i \quad i \quad eas \quad t \quad prove ’ ’ \quad an \quad ’ ’ ob vou s l y and standardfamous period book .. A large of P numberO lya of and such Szeg model proofso¨ 1. commaF − rom whichthis can book serve he as learned an what in .example . . Undoubtedly commaclassical can analysis b ate found that wasin tthe calledime book , of the around P acute-o mathematical 1 lya 925 and , Szeg there style o-dieresis : existed using mentioned important minimal above means and results t provedin it s references byo ingenious acquire period the and greatest beautiful possibility techniques of clear expression , which could and natural serve order as a in model proofs , andToday standard commawithout classical . \quad using analysisA phrases large has no such number such as importance of ” i such i and model eas attraction proofs t as prove it , had which ” an can ” serve ob vou as sly an example70 years ago ,. can . comma . Undoubtedlyb e in found the t ime in at when the that Karamata book t ime of , devoted P around $ \ allacute 1 his 925 days{o ,} there to$ this lya existed branch and important Szeg $ \ resultsddot{o} $ mentioned above and inof itmathematics s referencesproved period by ingenious Without . certain and beautifulbasic notions techniques of modern , mathematics which could comma serve such as a model and as .. the notionstandard .. of structure . A large comma number .. common of such .. properties model proofs .. of different , which .. can mathematical serve as an example Todayt opics , comma classical, can .. b classical e analysis found analysis in the has looks book no nowadays such of P importance foro´ lya many and mathematicians Szeg ando¨ attractionmentioned like .. a above as it and had in it s 70collection years ..references ago of i ,solated in . the .. propositions t ime when period Karamata .. If some devoted .. among .. all them his .. do days .. not to .. seethis .. the branch ofb eautymathematics of anToday i solated . Without , proposition classical certain analysis comma it basic has may no look notions such to them importance of as modernart for and the mathematics art attraction quoteright as s , it such had 70 assake\quad periodyearsthe .. But notion ago the , real in\quad results the tof in ime classical structure when mathematics Karamata , \quad and devotedcommon in every all\ branchquad his days ofp r it o p to e r this t i e s branch\quad of of different \quad mathematical twill o p i not c s disappear ,mathematics\quad inclassical the future . Without period analysis .. certain These looks are basic the nowadays resultsnotions which of for modern belong many neithermathematics mathematicians , such like as\quad a c ot l o l e some c t i o t nthe ime\quad notion commao fnot i of to s structure o some l a t e sp d ecial\ ,quad area commonpropositions hyphen these properties are . the\quad results ofI different whichf some open\quad the mathematicalamong \quad them \quad do \quad not \quad see \quad the bway eauty of progress oft opics an in i science , solated classical period proposition .. analysis new results looks ,comma it nowadays may new look consequences for to many them and mathematicians as new art theories for the like art ’ a s sakeoriginate . \quad fromcollection theseBut results the of real periodi solated results .. They propositions .. in are classical the stimulus . If ..mathematics somefor new aspirations among and periodin them every do branch not of it willD period not ..see disappear Hilb ert the .. b thought in eauty the .. of that future an i .. solated the . abundance\quad propositionThese and .. , are diversity it may the look of results important to them which .. as areas art belong for the neither art tof o mathematics some’ t s ime sake require , . not precise But to the some and real simple sp results ecial methods in area classical in resolving− these mathematics basic are and the and results in every which branch open of it the wayessential of progress problemswill not of disappear in these science areas in comma the . \quad future whichnew . then results These unite various are , the new ideas results consequences period which .. But belong the and neither new theories t o originatediscoveredsome .. from essence t imethese .. of, not these results to foundations some . sp\quad ecial comma areaThey .. -b\ these ecomequad are unexpectedlyare the the results stimulus new which .. areas\ openquad thefor way new of aspirations . D.having\quad theirprogressHilb own independent e r in t science\quad developmentthought . new results\ periodquad ..that , Onnew the\ consequencesquad otherthe hand abundance comma and new new simple theories and \quad originatediversity of important \quad areas of1 submathematics G periodfrom theseP requireacute-o results lya precise and . G They period and Szegsimple are the o-acute stimulus methods sub period in for Aufgabenresolving new aspirations und basic Lehrs . andacute-a D . tze Hilb aus der Analysis commaessential Springerert problems open thought parenthesis of these thatBerlin areas closing the abundance , parenthesis which then comma and unite diversity various of important ideas . \quad areasBut of the d i1 s 925c o v periode r e dmathematics\quad e s s e require n c e \quad preciseof and these simple foundations methods in , \ resolvingquad b ecome basic and unexpectedly essential new \quad areas having theirproblems own independent of these areas development , which then .unite\quad variousOn the ideas other . hand But the , new discovered simple essence of these foundations , b ecome unexpectedly new areas having their \ hspace ∗{\ownf i l l independent} $ 1 { G. development}$ P $ .\acute On the{o} other$ lya hand andG. , new simple Szeg $ \acute{o} { . }$ Aufgaben und Lehrs $ \acute{a} $1 tzeG. P auso´ lyader and G Analysis . Szeg o´. Aufgaben , Springer und Lehrs ( Berlina´ tze aus ) der , Analysis , Springer ( Berlin ) , 1 925 . \noindent 1 925 . 4 .... M period Tomi acute-c \noindentproofs which4 \ enlightenh f i l l M the . whole Tomi problem $ \acute comma{c} also$ give new results and bring important .. generalizations period .. Every .. discipline .. reaches .. its .. summit semicolon .. only .. it s \noindentsubstance remainsproofs period which enlighten the whole problem , also give new results and bring importantThe work of\ Jovanquad Karamatageneralizations b elongs to classical . \quad analysisEvery comma\quad whered i s cit i hasp l i n e \quad reaches \quad i t s \quad summit ; \quad only \quad i t s substancereached it s remains summit period . .. Karama had the power and the properties of the great analysts of this epoch period He loved elegance in proof comma which inspires enthusiasm Thefor work mathematics of4 Jovan comma Karamata and original b elongs proofs and to unexpected classical ideas analysis comma which , where are more it hasM . Tomi c´ reachedfrequent among itproofs s summit young which mathematicians .enlighten\quad Karama the period whole had .. He problem the shared power ,with also G and give period the new H properties period results Hardy andbring the of the impor- great analystsopinion thattant of mathematics this generalizations epoch i s a . young He loved . p eople Every elegance quoteright discipline sin game proof period reaches , .. which The b its inspires est three summit works enthusiasm ;of only forKaramata mathematicsit open s substance square , and bracket remains original 1 5 closing . proofs square and bracket unexpected comma open ideas square ,bracket which 1 6 are closing more square bracket comma openfrequent square bracket amongThe 1 young work 7 closing of mathematicians Jovan square Karamata bracket open . b parenthesis elongs\quad toHe classical the shared numbers analysis with refer G t , .o where theH . bibliography Hardyit has reached the of Karamata closing parenthesisopinion thatit s summit mathematics . Karama i s a had young the p power eople and ’ the s game properties . \quad of theThe great b est analysts three of works of Karamatab elongs tthis o[ the 1 epoch 5 t ime ] , of . [He his 1 lovedyouth 6 ] commaelegance , [ 1 and 7 in ] they proof ( the became , which numbers very inspires famous refer enthusiasmperiod t o the bibliography for mathematics of Karamata ) bAs elongs we said, t comma and o the original Karamata t ime proofs of has his foundand youthunexpected inspiration , and for ideas histhey thesis , which became in the are bookvery more of famous frequent . among young P acute-o lyamathematicians and Szeg o-dieresis . open He parenthesis shared with T period G . H I comma. Hardy section the 4opinion closing parenthesis that mathematics comma mentioned above periodAs we .. said His thesisi s , a Karamata young contains p a eople has found ’ s game inspiration . The b est for three his works thesis of in Karamata the book [ 1 of 5 ] , [ 1 6 Pgeneralization $ \acute] ,{ [o of 1} Weyl 7$ ] ( lyaquoteright the numbersand Szeg s to the refer $ power\ tddot oof the{ 2o notion bibliography} ( of $ uniform T of. distribution I Karamata , section period ) b 4 elongs ).. The , mentioned centralt o the prob above hyphen . \quad His thesis contains a lem studiedt imein his of thesis his youth i s the following , and they one became : very famous . \noindentLet open bracegeneralizationAs a wesub said nu n , closing Karamata of Weylbrace .. has ’ be $ found a boundeds ˆ{ inspiration2 comma}$ notion .. for double his of thesissequence uniform in of the real distribution book numbers of Pcommao´ . \ aquad less aThe sub central prob − nulem n less studied b commalya andin his Szeg thesiso¨( T . Ii , s section the following 4 ) , mentioned one : above . His thesis contains a 2 and .. let fgeneralization open parenthesis of x Weyl closing ’ parenthesiss notion .. of be uniform .. an R hyphendistribution integrable . function The central .. in .. prob open square - bracket a commaLet $ b\{ closinglema square studied{\ bracketnu in hisn semicolon}\} thesis$ i.. s find the\quad .. following conditionsbe a bounded one .. on : .. the , \quad double sequence of real numbers $ , a < a {\nu n } < b , $ s equence openLet brace{a aνn sub} nube n a closing bounded brace , and double on the function sequence f open of real parenthesis numbers x closing, a < parenthesis aνn < b, and .. that guarantee theand existence\quad ofletl the e tf(x $) fbe ( an xR )− $integrable\quad be function\quad an in $ R [a,− b];$find integrable conditions function \quad in \quad $ [following a limiton , b the ] ; $ \quad f i n d \quad c o n d i t i o n s \quad on \quad the limint n rights equence arrow infinity{aνn A} suband n on open the parenthesis function f closingf(x) parenthesisthat guarantee = limint the n right existence arrow infinity of the open brace 1 n sum\noindent from nu =followings 1 equence to n f open limit parenthesis $ \{ a a sub{\ nunu n closingn }\} parenthesis$ andonthefunction closing brace = A open parenthesis $f f ( closing x parenthesis )$ \quad that guarantee the existence of the following limit period ν=1 Karamata introduced first the notion of uniform distributionX of a double lim An(f) = lim {1n f(aνn)} = A(f). \ [ \lim { n \rightarrow \ninfty→∞ } An→∞{ n } ( f ) = \lim { n \rightarrow \ infty } sequence defined by n \{ nu open1 { parenthesisn }\sum x closingˆ{\nu parenthesis= 1= limint} { n n} rightf arrow ( infinity a {\ r subnu n nn open} parenthesis) \} x= closing A parenthesis ( f ).where\ ] r sub nKaramata open parenthesis introduced x closing first parenthesis the notion is the of number uniform of a subdistribution nu n less than of a x double comma for se- n fixed period .. For an ordinaryquence defined by sequence and if nu open parenthesis x closing parenthesis = x comma this reduces t o Weyl quoteright s problem of uniform Karamata introduced first the notion ofν(x uniform) = lim rnn distribution(x) of a double dis hyphen n→∞ sequencetribution of defined numbers bymodulo one period .. He finds that A open parenthesis f closing parenthesis exists whenever the where r (x) is the number of a less than x, for n fixed . For an ordinary distribution functionn nu open parenthesis x closingνn parenthesis .. exists comma .. for all except .. countably many points sequence and if ν(x) = x, this reduces t o Weyl ’ s problem of uniform dis - of\ [ \nu ( x ) = \lim { n \rightarrow \ infty } r { n{ n }} ( x ) \ ] tribution of numbers modulo one . He finds that A(f) exists whenever the open square bracket a comma b closing square bracket comma and excluding the discontinuities of nu open parenthesis x distribution function ν(x) exists , for all except countably many points of closing parenthesis and f open parenthesis x closing parenthesis period In this case [a, b], and excluding the discontinuities of ν(x) and f(x). In this case \noindentA open parenthesiswhere f$ closing r { parenthesisn } ( = x integral )$ sub is 0 to thenumberof the power of 1 f open $a square{\nu bracketn mu}$ open less parenthesis than xi $x ,$ for $n$ fixed . \quad For an ordinary closing parenthesis closing square bracket d xi comma Z 1 sequencewhere mu and open if parenthesis $ \nu x closing( x parenthesis )A =(f i) s = the x inversef ,[µ $(ξ)] ofdξ, this nu open reduces parenthesis t oWeyl x closing ’parenthesis s problem period of uniform He dis − givestribution several necessary of numbers and sufficient modulo one . \quad He0 finds that $A ( f ) $ exists whenever the distributionconditionswhere for the function existenceµ(x) i s the of $ nu inverse\nu open parenthesis( of ν x(x). He x ) closing $ gives\quad parenthesis severale x i necessary s tcomma s , \ generalizingquad and sufficientfor the all results except of Weyl\quad commacountably P many points of $ [ a , b ] , $ and excluding the discontinuities of $ \nu ( x ) $ and acute-o lia conditions for the existence of ν(x), generalizing the results of Weyl , P o´ lia 2H. $f2 sub ( H periodWeyl x Weyl ), U´ ber comma .$ die Gleichverteilung InthiscaseU-acute ber die von Gleichverteilung Zahlen mod . Eins von , Zahlen Math . mod Ann period. , Bd . Eins 77 ( comma 1 9 1 6 ) Math period Ann period comma Bd period 77 open parenthesis 1 9 1 6 closing parenthesis \ [ A ( f ) = \ int ˆ{ 1 } { 0 } f [ \mu ( \ xi ) ] d \ xi , \ ]

\noindent where $ \mu ( x )$ istheinverseof $ \nu ( x ) . $ He gives several necessary and sufficient

\noindent conditions for the existence of $ \nu ( x ) , $ generalizing the results of Weyl ,P $ \acute{o} $ l i a $ 2 { H. }$ Weyl $ , \acute{U} $ ber die Gleichverteilung von Zahlen mod . Eins , Math . Ann . , Bd . 77 ( 1 9 1 6 ) Jovan Karamata .... 5 \noindentand Fej acute-eJovan r period Karamata Using\ previoush f i l l results5 comma Karamata determines the domain of con hyphen vergence of the series .. sum a sub nu P sub nu open parenthesis x closing parenthesis comma .. where P sub nu open parenthesis\noindent xand closing Fej parenthesis $ \acute .. are{e polynomials} $ r . withUsing real previous results , Karamata determines the domain of con − vergencecoefficients of period the .... series This problem\quad reduces$ \sum to .... aa problem{\nu of} theP distribution{\nu } nu open( parenthesis x ) x , closing $ \quad parenthesiswhere $ Pof zeros{\nu of the} polynomial( x P ) sub $ nu\ openquad parenthesisare polynomials x closing parenthesis with real period In the case when P sub nu open parenthesis x closing parenthesis .... are orthogonal \noindentpolynomialsJovancoefficients comma Karamata the problem . \ h was f i l formulatedl This problem are solved reduces by Szeg acute-o to \ h to f i thel l a power problem of 3 sub of period the5 distribution $ \Unfortunatelynu (and x comma Fej )e´ $ ..r Karamata. Using previous quoteright results s thesis , comma Karamata .. published determines in Serbian the comma domain .. remained of con - un hyphen P known t overgence the majority of ofthe mathematicians series aν workingPν (x), inwhere the areaP ofν (x uniform) are polynomials with real \noindentdistributioncoefficientsof period zeros .. let of . us the mention This polynomial that problem a number reduces $ P of results{\ tonu from} a his( problem thesis x was of) the .$ distribution Inthecasewhenν(x) $P {\nu } ( x ) $ \ h f i l l are orthogonal proved independentlyof zeros of and the almost polynomial at the samePν (x time). In in the the case b eautiful when thesisPν (x of) are orthogonal 3 Schoenbergpolynomials to the power , of the 4 period problem was formulated are solved by Szeg o´. \noindentKaramatapolynomials publishedUnfortunately in Comptes , the , Rendus problem Karamata open was square ’ s formulated thesis bracket , 3 closing published are square solved in bracket Serbian by Szeg a theorem , $ remained\ relatedacute{ to o} linearˆ{ 3 } { . }$ functionalsun A sub - known n open parenthesist o the majority f closing ofparenthesis mathematicians comma which working he used in in the his thesis area periodof uniform .. It i s the following theoremUnfortunately : distribution , \quad .Karamata let us mention ’ s thesis that a , number\quad published of results from in Serbian his thesis , was\quad remained un − knownLet A subt oproved n the open majority parenthesis independently fof closing mathematicians and parenthesis almost ..at and the working A same open parenthesis time in the in the area f closing b eautiful of parenthesis uniform thesis .. of be .. linear functionals ..distribution of continuous functions . \quad .. onlet us mention that a number of results from his thesis was

open square bracket a comma b closing square bracket comma ..4 and let alpha sub n open parenthesis x closing parenthesis ..\noindent and alphaproved open parenthesis independently x closing parenthesisand almost ..Schoenberg atbe their the generating same. time functions in the comma b eautiful .. where thesis alpha sub of n open parenthesis a closing parenthesis = \ beginalpha{ opena l i g n parenthesis∗}Karamata a publishedclosing parenthesis in Comptes = 0 and Rendus alpha [ open 3 ] a parenthesis theorem related x closing t parenthesis o linear func- .. is nondecreasing semicolonSchoenberg thentionals in ˆ{ order4 A} thatn(f.) the, which sequence he used A sub in n hisopen thesis parenthesis . It f closing i s the parenthesis following theorem : \endconverge{ a l i g n ..∗} toLet .. A openAn(f parenthesis) and fA closing(f) be parenthesis linear .. for functionals .. every .. continuous of continuous function f open functions parenthesis x closing parenthesis ..on it .. is[a, .. b necessary], and let .. andαn(x) and α(x) be their generating functions , where Karamatasufficient published thatαn(a the) = sequenceα(a in) = Comptes alpha 0 and sub n Rendusα open(x) parenthesisis [ nondecreasing 3 ] a x closing theorem parenthesis ; then related in converges order t o thatlinear to alpha the open sequence parenthesis x closing parenthesisfunctionals inA everyn(f) $ continuityconverge A { n point} ( to fA( )f) for ,$ whichheused every continuous in his function thesis . \fquad(x) itIt i s the following theorem : of alpha openis parenthesis necessary x closing and parenthesis sufficient period that the sequence αn(x) converges to α(x) in LetJacques $ A Hadamard{everyn } continuity realized( f the point ) importance $ \quad ofand this simple $ A theorem ( f open ) parenthesis $ \quad seebe \quad linear functionals \quad of continuous functions \quad on $the [ same a Comptes , b Rendus ] immediately , $ \quad afterand Karamata l e t quoteright $ \alpha s note{ n closing} ( parenthesis x ) period $ \quad .. Thisand the hyphen $ \alpha (orem x can ) also $ be\quad foundbe in the their well hyphengenerating known functionsbook of Riesz , and\quad Nagywhere to the power $ \alpha of 5 open{ parenthesisn } ( page a ) ofα(x). = $1 2 1 closing parenthesis period $ \alpha ( a ) = 0 $ and $ \alpha ( x ) $ \quad is nondecreasing ; then in order that the sequence Karamata onceJacques said : .. Hadamard quotedblright realized .. I Wanted the t importance o improve my of knowledge this simple of the theorem foun hyphen ( see the $ Adations{ n of} the( theory f of )functions $ period .... This i s the reason why I first started t o converge same\quad Comptesto \quad Rendus$ A immediately ( f ) after $ \ Karamataquad f o r \ ’quad s noteevery ) .\ Thisquad thecontinuous - function study the theory of series comma but when I entered into this theory comma I found so 5 $ f (orem x ) can $ also\quad be foundi t \quad in thei s well\quad - knownn e c e s books a r y of\quad Rieszand and Nagy ( page 1 2 many old and1 ) .new results that I remained there period quotedblright .... In this area of analysis sufficienthe obtained beautiful that the and sequence important results $ \alpha comma which{ n } made( his x name )$ famous convergesto period $ \alpha ( x ) $ in everyKaramata continuity once point said : ” I Wanted t o improve my knowledge of the foun In the first- place these are his results about the summability comma and especially the proof of Littlewood quoteright s to the power of 6 theorem on the converse of Abel quoteright s theorem period .... \ begin { a l i gdations n ∗} of the theory of functions . This i s the reason why I first started t o This study the theory of series , but when I entered into this theory , I found so o f3 sub\ Galpha period Szeg( acute-o x ) sub comma . acute-U ber die Entwickelung einer analytischen Funktion nach den Polynomen \end{ a l i g n many∗} old and new results that I remained there . ” In this area of analysis eines Orthogonalsystemshe obtained beautiful comma Math and period important Ann period results 82 open , which parenthesis made his1 92 name 1 closing famous parenthesis . 1 88 hyphen 2 1 2 period Jacques HadamardIn the first realized place these the are importance his results of about this the simple summability theorem , and ( see especially 4 sub I periodthe proofSchoenberg of Littlewood comma U-dieresis ’ s6 theorem ber die asymptotische on the converse Verteilung of Abel reeller ’ s theoremZahlen mod . period This 1 comma Math periodthe sameZeit period Comptes comma Rendus immediately after Karamata ’ s note ) . \quad This the − 3 Szeg o´ U´ ber die Entwickelung einer analytischen Funktion nach den Polynomen eines Orthog- Bd period 2 8 openG. parenthesis, 1 928 closing parenthesis 1 71 hyphen 1 9 9 period \noindent onalsystemsorem can , also Math . be Ann found . 82 ( 1 in 92 1 the ) 1 88 well - 2 1− 2 . known book of Riesz and $ Nagy ˆ{ 5 } ( $ 5 sub Fr acute-e d e-acute ric¨ Riesz et B acute-e la Sz period hyphen Nagy comma Le c-cedilla ons d quoteright analyse page 4I. Schoenberg , U ber die asymptotische Verteilung reeller Zahlen mod . 1 , Math . Zeit . , Bd . fonctionnelle comma2 8 ( 1 928 Gautheir ) 1 71 - Villars 1 9 9 . comma 1 935 period 16 2 sub 1 J ) period . E period Littlewood comma The converse of Abel quoteright s theorem on power series comma proc period 5Fr´ed´eric Riesz et B e´ la Sz . - Nagy , Le c − cedilla ons d ’ analyse fonctionnelle , Gautheir London MathVillars period , 1 935 . \ hspaceSoc period∗{\ f open i l l }Karamata parenthesis 2 once closing said parenthesis : \quad comma’’ \ volquad periodI Wanted 9 open parenthesis t o improve 1 9 1 my 0 closing knowledge parenthesis of the 434 foun − 6 E . Littlewood , The converse of Abel ’ s theorem on power series , proc . London Math . Soc . hyphen 448 semicolonJ. G period H period Hardy and J period E period Littlewood comma Tauberian theorems \noindent (dations 2 ) , vol . 9 of ( 1 the 9 1 0 theory ) 434 - 448 of ; G functions . H . Hardy and . \ Jh . Ef i . l Littlewoodl This i , Tauberian s the reason theorems why I first started t o concerningconcerning power series power period series period . . . , Proc period . London comma Math Proc . Soc period . ( 2 London ) 1 3 ( 1 Math 9 1 4 ) period 1 74 - 1 Soc 9 1 period. open parenthesis 2 closing parenthesis 1 3 open parenthesis 1 9 1 4 closing parenthesis 1 74 hyphen 1 9 1 period \noindent study the theory of series , but when I entered into this theory , I found so

\noindent many old and new results that I remained there . ’’ \ h f i l l In this area of analysis

\noindent he obtained beautiful and important results , which made his name famous .

\noindent In the first place these are his results about the summability , and especially

\noindent the proof of Littlewood ’ $ s ˆ{ 6 }$ theorem on the converse of Abel ’ s theorem . \ h f i l l This

$ 3 { G. }$ Szeg $ \acute{o} { , }\acute{U} $ ber die Entwickelung einer analytischen Funktion nach den Polynomen eines Orthogonalsystems , Math . Ann . 82 ( 1 92 1 ) 1 88 − 2 1 2 .

$ 4 { I. }$ Schoenberg $ , \ddot{U} $ ber die asymptotische Verteilung reeller Zahlen mod . 1 , Math . Zeit . , Bd. 28 ( 1928 ) 171 − 1 9 9 .

\ hspace ∗{\ f i l l } $ 5 { Fr \acute{e} d \acute{e} r i c }$ Riesz et B $ \acute{e} $ l a Sz . − Nagy , Le $ c−cedilla $ ons d ’ analyse fonctionnelle , Gautheir Villars , 1 935 .

$ 6 { J. }$ E . Littlewood , The converse of Abel ’ s theorem on power series , proc . London Math . Soc . (2) , vol . 9(1910)434 − 448 ; G . H . Hardy and J . E . Littlewood , Tauberian theorems

\noindent concerning power series . . . , Proc . London Math . Soc . ( 2 ) 1 3 ( 1 9 1 4 ) 1 74 − 1 9 1 . 6 .... M period Tomi acute-c \noindenti s the classical6 \ h theorem f i l l M : . Tomi $ \acute{c} $ Suppose that f open parenthesis x closing parenthesis = sum a sub n x to the power of n is convergent for bar x bar less 1 and\noindent that f openi parenthesis s the classical x closing parenthesis theorem tends: to a finite limit s as x right arrow 1 minus 0 open parenthesis i period e period .. that the s eries sum a sub n is A hyphen summableSupposethat closing parenthesis $f ( semicolon x ) if = \sum a { n } x ˆ{ n }$ is convergent for $ \mid x a sub\mid n = O< open1$ parenthesis andthat 1 slash $f n closing ( parenthesis x )$ comma tends then the s eries sum a sub n .. converges and its sum isto s period a finite6 limit $ s $ as $x \rightarrow 1 − 0 ( $ i .M e . . Tomi\quadc´ that the s eries $ \Todaysum thereai s exists{ then classical} a$ large i s and theorem $ rich A lit− erature : $ summable on inverse )theorems ; if hyphen called $ a { n } = O ( 1 /P nn ) ,$ thentheseries $ \sum a { n }$ \quad converges and its sum is theorems of theSuppose Tauberian that typef period(x) = anx is convergent for | x |< 1 and that f(x) tends to $ s . $ P The first theorema finite which limit carriess as Tauberx → 1 quoteright− 0( i . e s . to the that power the of s7 eriesname opena parenthesisn is A− summable 1897 closing parenthesis .. P states that the) ; if an = O(1/n), then the s eries an converges and its sum is s. Todayconvergence there follows existsToday from there a Alarge hyphen exists and summability a rich large lit and open erature rich parenthesis lit eratureon and inverse from on inversesummability theorems theorems in− generalc a l l - e d called closing parenthesis theoremsunder an additionaltheorems of the Tauberian condition of the Tauberian comma type which . type is here . aa sub n right arrow 0 period A part of real anal hyphen 7 ysis b ears hisThe name first period theorem Littlewood which to the carries power of Tauber 6 open parenthesis ’ s name 19 ( 1 1897 0 closing ) parenthesis states that replaced the the Tauberian conditionThe firstconvergence theorem which follows carries from TauberA− summability ’ $ s ˆ{ (7 and}$ from name summability ( 1897 ) \ inquad generalstates ) that the convergence follows from $ A − $ summability ( and from summability in general ) aa sub n rightunder arrow an 0 additional by the condition condition na sub , nwhich = O open is here parenthesisaan → 0 1. closingA part parenthesis of real anal comma - and this condition is under an additional condition , which6 is here $ aa { n }\rightarrow 0 . $ Apart of real anal − so essential andysis b ears his name . Littlewood (1910) replaced the Tauberian condition aan → 0 importantby comma the condition that this branchnan = ofO analysis(1), and starts this with condition it period is ..so While essential the proof and of important , that \noindentthe Tauberianthisysis theorem branch b ears i of s veryanalysis his simple name starts open $ . with parenthesis Littlewood it . though Whileˆ theorem the{ 6 proof} i s( not of the at 19 all Tauberian trivial 1 0 closing theorem ) parenthesis $ replaced comma the Tauberian condition $the aa proof{ ofin s Littlewood}\ veryrightarrow simple quoteright ( though s0 theorem $theorem by is the not i s onlycondition not complicated at all trivial $ comma na ) ,{ the butn } also proof= involved of O Littlewood ( 1 ’ ) , $ and this condition is so essential and importantand very longs ,theorem period that Hardy this is not andbranch only Littlewood complicated of analysis to the power , but starts of also 6 open involved with parenthesis it . \quad 1 9 13 closingWhile parenthesis the proof gave of a new proof 6 usingthe auxil Tauberian hyphenand very theorem long . i Hardy s very and simpleLittlewood ( though(1913) theoremgave a new i s proof not at using all auxil trivial - iary ) , theiary proof theoremstheorems of connected Littlewood connected with the ’ swith relations theorem the b relations etween is not real b etweenonly functions complicated real and functions their , and but their also derivatives involved derivatives. period This .. This proof proof , including comma including the auxiliary the auxiliary results results , has comma several has pages several and pages also and some \noindentalso somereferences and very t ot long previouso previous . Hardy theorems theorems and period $ . Littlewood.. Still Still comma , their ˆtheir{ proof6 proof} uses( uses only only1 simple 9 13 ) $ gaveanewproof using auxil − iarymethods theorems ofmethods analysis connected period of analysis In 1 with 925 . comma In the 1 925 relations Schmidt , Schmidt 8 gave b etween the8 gave proofthe real of this proof functions theorem of this usingtheorem and their using derivativesnew notionsnew and . notions auxiliary\quad andThis theorems auxiliary proof comma , theorems including but his proof , but the was his neither auxiliary proof simple was neither results nor simple , has nor several pages and alsoshort some periodshort references In his . monographIn his t monograph o comma previous Landau , Landau theorems 9 says9 thatsays . Littlewood\quad that LittlewoodStill quoteright , their s’ theorems theorem proof is a uses is very a very only simple deep one commadeep one but , the but proof the i proof s complicated i s complicated period .. then . Karamata then Karamata open square [ 1 6bracket ] published 1 6 closing his square bracket published\noindent his spmethods ectacular of proof analysis of two . pages In 1 925. , Schmidt 8 gave the proof of this theorem using newsp ectacular notions proofThe and of elementary auxiliary two pages period Karamatatheorems , method but his is proofbased onwas the neither Weierstrass simple theorem nor on The elementaryapproximation Karamata method by polynomials is based on . the Weierstrass If we put theorem \noindenton approximationshort by . Inpolynomials his monograph period .. If , we Landau put 9 says that Littlewood ’ s theorem is a very deep one , but the proof i s complicated, . \equad then1 Karamata [ 1 6 ] published his g open parenthesis x closing parenthesisg =(x open) = { brace01, 01 10 sub≤ ≤ commaxx ≤ ≤e to1 the, power of comma 1 from e to 0 less or equal lesssp or ectacular equal x sub x proof less or of equal two less pages or equal . e from 1 to 1 comma 7A. Tauber , Ein Satz aus der Theorie der unendlichen Reihen . Monatshefte f u¨ r Math . 8 ( 1 7 sub A period Tauber comma Ein Satz aus der Theorie der unendlichen Reihen period Monatshefte f u-dieresis r Math The elementary897 ) , 273 Karamata - 277 . method is based on the Weierstrass theorem period ¨ on approximation8R. Schmidt by polynomials, U ber das Borelsche . \quad SummierungsverfahrenI f we put , Schriften der K o¨ nigsberger gelehrten 8 open parenthesisGesellschaft 1 897 , I (1925)202 closing parenthesis− 256; U¨ commaber divergente 273 hyphen Folgen 277 und period lineare Mittelbildun - gen , M . Zeit . , 8 sub R period22 ( 1 9Schmidt 25 ) 89 - comma 1 52 . U-dieresis ber das Borelsche Summierungsverfahren comma Schriften der K o-dieresis nigsberger\ [ g ( x ) = \{ 0 { 1 }ˆ{ , } { , } 1 ˆ{ e } { 0 }\ leq {\ leq } x { x }\ leq {\ leq } 9E. Landau , Darsellung und Begr u¨ ndung einiger neurer Ergebnisse der Funktionen - theorie , e ˆgelehrten{ 1 } { Gesellschaft1 } , \ ] comma I open parenthesis 1 92 5 closing parenthesis 202 hyphen 256 semicolon U-dieresis ber Springer , Berlin ( 1 930 ) ; Bull . Acad . Belgique 1 9 1 1 , see P o´ lya and Szeg o¨, vol . I , pp . 231 , divergente Folgenwhere und the lineare term ” Mittelbildunlangsa−m hyphenwachsend Funktio n a w el a th notatio L(x) and gen comma M period Zeit period comma 22 open parenthesis 1 9 25 closing parenthesis 89 hyphen 1 52 period $9 7 sub{ EA. period Landau}$ Tauber comma , Darsellung Ein Satz und aus Begr der u-dieresis Theorie ndung der einiger unendlichen neurer Ergebnisse Reihen der .Funktionen Monatshefte hyphen f $ \ddot{u} $ r Math . theorie comma Springer comma BerlinL open(cr)/L parenthesis(r) → 1, c > 11 930, areintroduced closing parenthesis. semicolon Bull period Acad period Belgique8 ( 1 1 897 9 1 1 ) comma , 273 see− P277 acute-o . lya and Szeg o-dieresis sub comma vol period I comma pp period 231 comma where the term quotedblright .. langs to the power of a-m .. wachsend .. Funktio n .. a w el .. a .. th$ .. 8 notatio{ R. L open}$ parenthesis Schmidt x closing $ , parenthesis\ddot{U and} $ ber das Borelsche Summierungsverfahren , Schriften der K $ \Lddot open{o parenthesis} $ nigsberger cr closing parenthesis slash L open parenthesis r closing parenthesis right arrow 1 comma c greater 1 commagelehrten are introduced Gesellschaft period , I $( 1 92 5 ) 202 − 256 ; \ddot{U} $ ber divergente Folgen und lineare Mittelbildun − gen ,M. Zeit . , 22 ( 1925 ) 89 − 1 52 .

$ 9 { E. }$ Landau , Darsellung und Begr $ \ddot{u} $ ndung einiger neurer Ergebnisse der Funktionen − theorie , Springer , Berlin ( 1 930 ) ; Bull . Acad . Belgique 1 9 1 1 , see P $ \acute{o} $ lya and Szeg $ \ddot{o} { , }$ vol . I , pp . 231 , where the term ’’ \quad $ langs ˆ{ a−m }$ \quad wachsend \quad Funktio n \quad a w e l \quad a \quad th \quad n o t a t i o $ L ( x ) $ and

\ begin { a l i g n ∗} L ( cr ) / L ( r ) \rightarrow 1 , c > 1 , are introduced . \end{ a l i g n ∗} Jovan Karamata .. 7 \noindentthen a polynomialJovan Karamata p open parenthesis\quad x7 closing parenthesis such that thenintegral a polynomial sub 0 to the power $p of 1 ( bar x x to the ) $ power such of minus that 1 g open parenthesis x closing parenthesis minus p open parenthesis x closing parenthesis bar dx less epsilon \ [ corresp\ int ˆ onds{ 1 t} o every{ 0 }\ epsilonmid greaterx 0 ˆ period{ − The1 } proofg i s obvious ( x for ) the power− p functions ( x x to the ) power\mid of n commadx < \ varepsilonand for the polynomial\ ] the proof is carried by using approximation period A survey of all methods used in the proofs of Littlewood quoteright s theorem can be found in theJovan article Karamata of Ingham7 to then the power a polynomial of 10 periodp( ..x) Amongsuch that the elementary ones comma Karamata quoteright s \noindentproof standscorresp .. out comma onds .. t and o everyit .. is .. cited$ \ varepsilon as .. a model and> .. a0 standard . $ in The .. all proofwell hyphen i s obvious for the power functions $ x ˆ{ n } , $ Z 1 known monographs in this field comma for example| x−1 ing(x those) − p by(x) E| dx period < ε C period Titchmarsh comma and for the polynomial the proof is carried by using approximation . G period .. H period .. Hardy period .. K period0 .. Knopp period .. A period .. Zygmund comma .. D period .. V period .. Widder comma .. G period .. Doetsh periodε > 0. .. The xn, \ hspace ∗{\correspf i l l }A onds survey t o everyof all methodsThe proof used i in s obvious the proofs for the of power Littlewood functions ’ s theoremand can be importancefor of the Littlewood polynomial quoteright the s proof theorem is carried li es not by only using in the approximation fact that it i s one . of the crucial inverse theorems comma but that it is also an important result in \noindent foundA in survey the of article all methods of $ used Ingham in the ˆ{ proofs10 } of. Littlewood $ \quad ’Among s theorem the can elementary be ones , Karamata ’ s the Theoryfound of F-u in sub the nctions article period of Ingham .. R period10. Among Schmidt thecomma elementary in his review ones in , the Karamata Zentralblatt ’ s comma proof says proofthat Karama stands quoteright\quad out s proof , \ iquad s quotedblrightand i t ..\quad a .. importani s \quad .. dc iscover i t e d y as quotedblright\quad a model .. K .. and Knop\ toquad the powera standard in \quad a l l w e l l − known monographsstands inout this , and field it , is for cited example as in a model those and by E a . standard C . Titchmarsh in all well , of 1 1 .. says that- known monographs in this field , for example in those by E . C . Titchmarsh , G G.this\ iquad s a simpleH. comma\quad ..Hardy surprising . \quad proof openK. parenthesis\quad Knopp Ein u-acute . \quad beraschendA. \quad BeweisZygmund closing parenthesis , \quad D. period\quad .. V. \quad Widder , \quad G. \quad Doetsh . \quad The importance. of H Littlewood . Hardy . ’ s K theorem . Knopp li . es not A . only Zygmund in the , fact D . that V it. i Widder s one In .. 1 978 comma, G .. on . Doetsh . The importance of Littlewood ’ s theorem li es not only ofthe the occasion crucial of the inverse 60 th anniversary theorems of the , Mathematische but that it Zeitschrift is also comma an important this result in the Theoryin of the fact $ F− thatu { itn i ct s i oone n s of . the}$ crucial\quad inverseR . Schmidt theorems , , in but his that review it is also in anthe Zentralblatt , says work of Karamataimportant quoteright result s in was the chosen Theory among of theF − firstunctions 50 most. R important . Schmidt papers , in his review in the selected between several hundreds of papers period \noindent Zentralblattthat Karama , says ’ s proof i s ’’ \quad a \quad importan \quad d iscover y ’’ \quad K \quad F-r sub om Littlewood quoteright s theorem one can deduce the order of the remainder Knop11 $ Knop ˆ{ that1 1 Karama}$ \quad ’ s proofsays i s that ” a importan d iscover y ” K says bar s sub nthat minus this s bar i s = a osimple open parenthesis , surprising 1 slash proof lg n closing ( Ein parenthesisu´ beraschend comma Beweis and this ) . was In done 1 by Postnikov to thethis power i of s 1 a 2 simple and finally , by\quad F-r eudsurprising to the power proof of 1 3 ( Ein $ \acute{u} $ beraschend Beweis ) . \quad In \quad 1 978 , \quad on the occasion978 , of on the the 60 occasion th anniversary of the 60 ofth anniversary the Mathematische of the Mathematische Zeitschrift Zeitschrift , this and Korevaar, this to the work power of Karamata of 14 period ’ .... s was The chosenremainder among i s obtained the first essentially 50 most from important the Weierstrass papers workapproximation of Karamata theorem ’ commas was i chosen period e amongperiod from the it first s generalization 50 most given important by Jackson papers to the power of 1 5 period It i s selectedselected between between several several hundreds hundreds of papers of papers . . obtained by using classicalF − rom approximationLittlewood ’ theorems s theorem open one parenthesis can deduce Weierstrass the order and of Jack the hyphen remainder son closing| s parenthesis− s |= o(1 with/ an help), of a method similar t o KaramataPostnikov12 quoteright s semicolonF − andreud b13 ecause of that it is \ hspace ∗{\ f in l l } $ F−r lg{ omand}$ this Littlewood was done by’ s theorem oneand can finally deduce by the order of the remainder sometimesand calledKorevaar Karamata14. quoterightThe s remainderelementary method i s obtained period ..essentially This order from of the the remain Weierstrass hyphen der term iapproximation s used in number theorem theory period , i . This e . addition from it to s the generalization Littlewood theorem given comma by Jackson15. It i s \noindent1 0 sub A period$ \mid E periods Ingham{ n } comma − Procs period\mid London= Math o period ( 1 Soc period /$ 14 lg a open $n parenthesis ) ,$ 1 965 andthiswasdoneby closing $ Postnikovobtained ˆ{ 1 by 2 using}$ and classical finally approximation by $F−r theorems eud ˆ{ (1 Weierstrass 3 }$ and Jack - son parenthesis 1) 57 with hyphen a help1 73 period of a method similar t o Karamata ’ s ; and b ecause of that it is 1 1 sub K period Knopp comma Theorie und Anwendung der Unendlichen Reihen comma Dritte Auflage comma Springer \noindent sometimesand $ Korevaar called Karamata ˆ{ 14 } ’. s elementary $ \ h f i l l methodThe remainder . This i order s obtained of the remain essentially - from the Weierstrass comma der term i s used in number theory . This addition to the Littlewood theorem , berlin comma 1 93 1 period10 E . Ingham , Proc . London Math . Soc . 14 a ( 1 965 ) 1 57 - 1 73 . \noindent approximationA. theorem , i . e . from it s generalization given by $ Jackson ˆ{ 1 1 2 sub A period11K. GKnopp period , TheoriePostnikov und comma Anwendung The remainder der Unendlichen term Reihenin the Tauberian , Dritte Auflage theorem , Springer of Hardy , berlin and Little hyphen 5 }wood.comma $ I t Doklady i s Akad period Nauk period SSSR open parenthesis N period S closing parenthesis comma 77 open obtained, by 1 93 using 1 . classical approximation theorems ( Weierstrass and Jack − parenthesis 1 95 112 closingA. G . Postnikovparenthesis , The comma remainder 1 93 hyphen term in 1 the 96 Tauberian period theorem of Hardy and Little - wood , son1 3 ) sub with G period a help Freud of comma a method Restglied similar eines Tauberschen t o Karamata Satzes ’ I scomma ; and Acta b periodecause Math ofperiod that itAcad is period Sci sometimesDoklady called Akad Karamata . Nauk . SSSR ’ s ( elementary N . S ) , 77 ( 1 method 95 1 ) , 1 93 . -\quad 1 96 . This order of the remain − period Hung period13 2Freud open parenthesis , Restglied eines 1 95 Tauberschen 1 closing parenthesis Satzes I , Acta . Math . Acad . Sci . Hung . 2 ( 1 95 1 ) der term i s usedG. in number theory . This addition to the Littlewood theorem , 2 99 hyphen2 99 308 - 308 period . 14 sub J period Korevaar14 commaKorevaar Best , Best L subL 1Approximation Approximation and and the theRemainder Remainder in Littlewood in Littlewood ’ s Theorem quoteright , s Theorem \ centerline { $ 1 0 J.{ A. }$ E1 . Ingham , Proc . London Math . Soc . 14 a ( 1 965 ) 1 57 − 1 73 . } comma Konikl . Nederl . Akademie van Wetenschappen , Amsterdam , A . 56 , 3 ; Indag . Math . 1 5 ( 3 ) Konikl period Nederl period Akademie van Wetenschappen comma Amsterdam comma A period 56 comma 3 semicolon $ 1 1 {( 1K. 953 ) 281}$ - 293 Knopp . , Theorie und Anwendung der Unendlichen Reihen , Dritte Auflage , Springer , Indag period Math15 periodJackson 1 5, openU¨ ber parenthesis die Genauigkeit 3 closing der Ann parenthesisa¨ herung stetiger Funktionen durch ganze ra - tionale berlin , 1 93D 1. . open parenthesisFunktionen 1 953 gegebenen closing parenthesis Grades und 281trigonometrische hyphen 293 Summen period gegebener Ordnung 1 5 sub D period( Diss . Jackson G o¨ ttingen comma 1 9 U-dieresis 1 1 ) ; The ber theory die ofGenauigkeit approximation der Ann, 1 930 a-dieresis . herung stetiger Funktionen durch ganze ra$ hyphen 1 2 { A. }$ G . Postnikov , The remainder term in the Tauberian theorem of Hardy and Little − woodtionale , FunktionenDoklady Akad gegebenen . Nauk Grades . SSSR und trigonometrische ( N . S ) , Summen 77 ( 1 gegebener 95 1 ) Ordnung , 1 93 − 1 96 . open parenthesis Diss period G dieresis-o ttingen 1 9 1 1 closing parenthesis semicolon The theory of approximation comma 1$ 930 1 period 3 { G. }$ Freud , Restglied eines Tauberschen Satzes I , Acta . Math . Acad . Sci . Hung . 2 ( 1 95 1 ) 2 99 − 308 .

\ hspace ∗{\ f i l l } $ 14 { J. }$ Korevaar , Best $ L { 1 }$ Approximation and the Remainder in Littlewood ’ s Theorem ,

\noindent Konikl . Nederl . Akademie van Wetenschappen , Amsterdam , A . 56 , 3 ; Indag . Math . 1 5 ( 3 )

\noindent ( 1 953 ) 281 − 293 .

$ 1 5 { D. }$ Jackson $ , \ddot{U} $ ber die Genauigkeit der Ann $ \ddot{a} $ herung stetiger Funktionen durch ganze ra − tionale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung

\noindent ( Diss . G $ \ddot{o} $ ttingen 1 9 1 1 ) ; The theory of approximation , 1 930 . 8 .... M period Tomi acute-c \noindentwhich i s numerically8 \ h f i l l importantM . Tomi comma $ \acute .... follows{c} from$ elementary procedures comma .... that i s from Karamata quoteright s proof period Wielandt to the power of 1 6 showed that one can avoid the step \noindentin the proofwhich of Littlewood i s numerically quoteright s theorem important using C , hyphen\ h f i l l summabilityfollows from semicolon elementary that i s comma procedures one can , \ h f i l l that avoid Hardy quoteright s lemma and every auxiliary theorem in the proof comma except for the \noindentWeierstrassi approximation s from Karamata theorem ’ period s proof .. Let us $ note . that Wielandt the proof ˆ{ of1 Hardy 6 quoteright}$ showed s that one can avoid the step inlemma the is proof t oday of quite Littlewood simple period ’ s theorem using $ C − $ summability ; that i s , one can avoidThe complete Hardy8 survey ’ s lemma of Tauberian and every theorems auxiliary with remainder theorem was given in the proof , exceptM for . Tomi thec´ Weierstrassby Ganeliuswhich 17 approximation i s numerically theorem important . \quad , followsLet us from note elementary that the procedures proof of , Hardy that ’ s 16 lemmaThe big is andi t svery oday from important quite Karamata simple treatise ’ s of proof . Wiener. toWielandt the powershowed of 18 appeared that one in 1 can 932 period avoid .. the Us step hyphen in ing Fourierthe transforms proof of comma Littlewood Wiener gave ’ s theorem the most general using theoryC− summability of Tauberian ; that i s , one can \ hspacetheorems∗{\ periodavoidf i l l } ..The Hardy Wiener complete ’ quoteright s lemma survey s and theory every of not Tauberian auxiliaryonly includes theorem theorems the most in important with the proof remainder forms , except of was for given the general inverseWeierstrass theorems approximation comma .. but also theorem contains new. results Let us outside note this that area the comma proof of Hardy ’ s \noindentsome of whichlemmabyGanelius are the is t deepest oday 17 quite theorems simple in analysis . comma .. having a wide range of application comma fromThe integral complete equations survey t oof the Tauberian distribution theorems of primes with open remainderparenthesis S was period given Ike hyphen Thehara big closing andby parenthesis very Ganelius important period17 .. treatiseEvery analyst of can $ feel Wiener the depth ˆ{ of18 the}$ ideas appeared and of the in techniques 1 932 . \quad Us − 18 ingof many Fourier resultsThe transforms that big were and scattered very , Wiener important b efore gave Wiener treatise the period most of ..Wiener But general the synthesisappeared theory and in of 1 Tauberian 932 . Us - ing theoremsthe essenceFourier . comma\quad transforms andWiener also the ’ structure, s Wiener theory of gave the not whole the only most problem includes general comma theory the canmost b of e understood Tauberian important theorems forms of . generalonly by studying inverseWiener Wiener ’ theorems s theory quoteright not , \quad s only theory includesbut period also .. the This contains most theory important is new based results on forms Fourier outside of trans general hyphen this inverse area , someforms of and whichtheorems convolutions are , the period but deepest If also we denote contains theorems by f-macron new in results analysis the Fourier outside , transform\ thisquad areahaving of f , in some L sub a wide of 1 comma which range are ofi period application ethe period deepest , from theorems integral in analysis equations , t having o the a distribution wide range of of application primes ( , S from . Ike − open parenthesisintegral 2 pi equations closing parenthesis t o the distribution to the power of minus primes 1period ( S . Ike 2 minus - to the power of infinity infinity f open parenthesis\noindent xhara closing ) ) parenthesis . . \ Everyquad eEvery analystto the power analyst can of feel iux can the dx comma depth feel the of the depth ideas of and the of the ideas techniques and of the of techniques ofand many by K-macron resultsmany results sub that 1 .. thethat were Fourier were scattered transform scattered b of before K efore sub 1 Wiener commaWiener .. . . then\quad comma ButBut the .. ifthe synthesis the synthesiscondition and macron-K the and sub 1 open parenthesisthe essence uessence closing , and parenthesis , and also also theequal-negationslash the structure structure of of the the whole whole problem problem , can , can b e b understood e understood only only0 holds by for studyingby every studying real Wiener u comma Wiener ’ .. s we ’ s theory arrive theory to . .the\quad special ThisThis Wiener theory theory theorem is based is of on thebased Fourier on Fouriertrans - forms trans − forms and convolutions . If we denote by¯ $ \bar{ f } $ the Fourier transform of $ f \ in Tauberianand type convolutions which says : . If we denote by f the Fourier transform of f ∈ L1, L If{ 1 } i, . $ e . limint x right arrow infinity integral K sub 1 open parenthesis x minus y closing parenthesis a open parenthesis y closing −1.2 ∞ iux parenthesis\noindent dyi = . A e integral . K sub 1 open parenthesis(2π) y closing− ∞f parenthesis(x)e dx, dy comma where K-macron sub 1 open parenthesis u closing parenthesis negationslash-equal 0 for every real u comma K sub 1 in L and by K¯1 the Fourier transform of K1, then , if the condition K¯1(u) 6= 0 sub\ [1 ( comma 2 then\ pi ) ˆ{ − 1 . 2 } − ˆ{\ infty }\ infty f ( x ) e ˆ{ iux } dx , \ ] holds for every real u, we arrive to the special Wiener theorem of the Tauberian limint y righttype arrow which infinity says a : open parenthesis y closing parenthesis = A comma provided that a open parenthesis y closing parenthesis belongsIf to the class of functions satisfying the Tauberian conditions in a precisely defined way stated below period Today such functions are \noindentcalled s lowlyand oscillating by $ \ functionsbar{K} period{ 1 ..}Z$ The\quad approximationthe Fourier byZ the functions transform of of $ K { 1 } , $ \quad then , \quad if the condition $ \bar{K} { 1 } ( u ) \limne $ K1(x − y)a(y)dy = A K1(y)dy, 1 6 sub N period Wielandt comma Zurx→∞ Umkehrung des Abelschen Stetigkeitsatzes comma Math period Zeit period 56 comma0 holds open for parenthesis every 1 real 9 52 closing $u parenthesis , $ \quad we arrive to the special Wiener theorem of the where K¯ (u) 6= 0 for every real u, K ∈ L , then Tauberian206 hyphen type 207 period which1 says : 1 1 1 7 sub T period Ganelius comma Tauberian Remainder Theorems comma Lecture Notes in Math period comma vol period \ centerline { I f } lim a(y) = A, 1 31 comma 1 971 period y→∞ 18 sub N period Wiener comma Tauberian Theorems comma Annals of mathematics comma Vol period 33 comma No provided that a(y) belongs to the class of functions satisfying the Tauberian con- period\ [ \lim 1 open{ parenthesisx \rightarrow 1 932 closing\ parenthesisinfty }\ 1 hyphenint 1K 0 period{ 1 } ( x − y ) a ( y ) dy = Aditions\ int in aK precisely{ 1 } defined( y way ) stated dy below , \ ] . Today such functions are called s lowly oscillating functions . The approximation by the functions of 16N. Wielandt , Zur Umkehrung des Abelschen Stetigkeitsatzes , Math . Zeit . 56 , ( 1 9 52 ) 206 - \noindent 207where . $ \bar{K} { 1 } ( u ) \not= 0$ foreveryreal $u , K { 1 } \ in L { 1 } 17,T. $Ganelius then , Tauberian Remainder Theorems , Lecture Notes in Math . , vol . 1 31 , 1 971 . 18N. Wiener , Tauberian Theorems , Annals of mathematics , Vol . 33 , No . 1 ( 1 932 ) 1 - 1 0 . \ [ \lim { y \rightarrow \ infty } a ( y ) = A , \ ]

\noindent provided that $ a ( y ) $ belongs to the class of functions satisfying the Tauberian conditions in a precisely defined way stated below . Today such functions are called s lowly oscillating functions . \quad The approximation by the functions of

$ 1 6 { N. }$ Wielandt , Zur Umkehrung des Abelschen Stetigkeitsatzes , Math . Zeit . 56 , ( 1 9 52 ) 206 − 207 .

\ hspace ∗{\ f i l l } $ 1 7 { T. }$ Ganelius , Tauberian Remainder Theorems , Lecture Notes in Math . , vol . 1 31 , 1 971 .

\ hspace ∗{\ f i l l } $ 18 { N. }$ Wiener , Tauberian Theorems , Annals of mathematics , Vol . 33 , No . 1 ( 1 932 ) 1 − 1 0 . Jovan Karamata .... 9 \noindentthe subspaceJovan generated Karamata by the\ translationsh f i l l 9 of K sub 1 plays a fundamental role in the whole of Wiener quoteright s theory : .. if K sub 1 in L sub 1 comma K in L sub 1 comma K-macron sub 1 open parenthesis\noindent u closingthe subspace parenthesis generated equal-negationslash by the 0 translationsfor every of $ K { 1 }$ plays a fundamental role inreal the u comma whole then of t Wiener o every epsilon ’ s theory greater equal : \quad 0 commai f there $ K corresponds{ 1 }\ ain sequenceL { A sub1 } n and,K a sequence\ in L { 1 } , of\ realsbar{ xiK} sub{ n comma1 } ( such u that ) \ne 0 $ f o r every realintegral $u bar K open,$ parenthesis thentoevery x closing parenthesis $ \ varepsilon minus sum A\geq sub n K0 sub 1 , open $ thereparenthesis corresponds x plus xi sub a n sequenceclosing parenthesis$ A { n bar}Jovan$ dx and less Karamata ora equalsequence epsilon comma 9 o f r e a l s $ \ xi { n } , $ such that and K-macronthe subspacesub 1 open generated parenthesis uby closing the translations parenthesis equal-negationslash of K1 plays a fundamental 0 i s the necessary role and in thesufficient condition ¯ for this approximawhole hyphen of Wiener ’ s theory : if K1 ∈ L1,K ∈ L1, K1(u) 6= 0 for every real u, then \ [ \ int \mid K ( x ) − \sum A { n } K { 1 } ( x + \ xi { n } ) t ion periodt o .. every This theoremε ≥ 0, there was very corresponds important in a the sequence creationA ofn abstractand a sequence harmonic of reals ξn, such \midanalysisdx commathat\ leq and also\ varepsilon in the development, \ of] some modern branches of contem hyphen porary mathematics period Z X In the scope of Wiener quoteright s theory| K one(x) finds− theAn greatestK1(x + ξ numbern) | dx ≤ ofε, inverse \noindenttheorems ofand the general $ \bar form{K} comma{ 1 ..} even( those u which ) at\ne the first0 glance $ i do s thenot necessary and sufficient condition for this approxima − t ion . \quad This theorem was very important in the creation of abstract harmonic appear t oand be ofK Tauberian¯1(u) 6= 0 naturei s the period necessary .. But numerical and sufficient problems condition remain out for of this approxima - t analysisthe reach ofion , harmonic and . also This analysis in theorem the period development ..was Special very problems important of some : .. the modern in connectionsthe creation branches between ofof abstract contem harmonic− porarythe relationships mathematicsanalysis of different , and . also summability in the development procedures comma of some estimation modern of branches the re hyphen of contem - porary mainder commamathematics asymptotic . expansion of the sum s sub n comma cannot be seen in the general Intheory the scope periodIn .. of Here the Wiener the scope breadth ’ of s Wienertheory of general one’ harmonic s theory finds analysis one the finds greatest can b the e seen greatest number in the ofnumber inverse of inverse theoremsgeneral outlooktheorems of the on the general of whole the problem general form , including form\quad , aeven number even those of those scattered which which re at hyphen at the the first first glance glance do do not not appearsult s comma tappear o be .. while of t o Tauberian classical be of Tauberian analysis nature reveals nature . the\quad inner.But But structure numerical numerical of a problem problems problems by remain remain out out of ofthe theusing reach sp ecialreach of means harmonic of harmonic comma analysis .. by analysis a breakthrough . .\quad Special intoSpecial the problems core problems of a : single the i : solated connections\quad the connections between the between theproblem relationships periodrelationships of of different different summability summability procedures procedures , estimation , estimation of the reof - the mainder re − , mainder , asymptotic expansion of the sum $ s { n } , $ cannot be seen in the general Wiener quoterightasymptotic s theory expansion was presented of the by sum Pitts ton, cannot the power be of seen 1 9 comma in the general1 938 comma theory .. in . a most Here precise and theoryconcise way. the\quad period breadthHere In order of the generalt o breadth arrive harmonic t o Karamataof general analysis quoteright harmonic can s b result e seenanalysis in this in thetheory can general comma b e seen outlook through in onthe generalFourier transforms outlookthe whole comma on problem the .. wholeone including should problem take a the number including transform of scattered of a f opennumber parenthesis re -of sult scattered x s closing , while parenthesis re − classical = e to the power ofs x u el t to s the , poweranalysis\quad ofwhile e reveals to the classical power the inner of x comma structure analysis with of reveals a problem the by inner using structure sp ecial means of a , problem by a by usingthe translation spbreakthrough ecial of the means form into , ..\ fquad open the parenthesiscoreby a of breakthrough a single x plus i lg solated mu closing into problem the parenthesis core . periodof a ..single Even with i solated this comma .. it i s notproblem .. easy to . Wiener ’ s theory was presented by Pitt19, 1938, in a most precise and concise arrive t o theway result . In without order t long o arrive calculations t o Karamata period .. It ’ iss resultnot easy in comma this theory .. b ecause , through the x Wienerauxiliary ’ s propositionsFourier theory transforms was in the presented general , theory one by should require $ Pitt take many ˆ the{ explanations1 transform 9 } period, of f 1 ..(x This) 938 = exe ,e $, with\quad thein a most precise and concisei s why Karamata waytranslation . In quoteright order of the ts proof formo arrive remainsf(x t+ classical olg Karamataµ) comma. Even and ’ s with this result i this s why , in Wiener this it i sayss theory not : .. quotedblright easy , through to .. th elegant methodarrive of t Karamata o the result open without parenthesis long Wiener calculations to the power . of It18 iscomma not easy p period , 5 b 1 ecauseclosing parenthesis the requires only\noindent one t extbookauxiliaryFourier propositionstransforms in , \ thequad generalone should theory require take the many transform explanations of . $ f This ( i x ) = e ˆtheorem{ x } :se .. why the ˆ{ Weierstrasse Karamata ˆ{ x }} approximation ’ s, proof $ with remains Case 1 quotedblright classical , and Case this 2 period i s why Wiener says : ” theThe translation abstractth version of of Wiener the form quoteright\quad s theory$ f was ( given x by + Beurling $ l g to the$ \mu power of) 20 period . $ Among\quad Even with this , \quad i t i s not \quad easy to arrivethe others telegant o who the have result method spread thesewithout of Karamata abstract long ideas calculations(Wiener one should18, p mention . . 5\ 1quad the ) requiresIt is onlynot easy one t , extbook\quad b ecause the auxiliarynames of I period propositions Gelfand comma in the R period general Godement theory and Srequire period L many period00 explanations Segal period . \quad This i s why Karamatatheorem : ’ s the proof Weierstrass remains approximation classical , andtheorem this i s why Wiener says : \quad ’’ \quad th 1 9 sub H period R period Pitt comma General Tauberian Theorems comma. Proc period London Math period Soc period comma vol period 44 open parenthesis 1 938 closing parenthesis 20 The abstract version of Wiener ’ s theory was given by Beurling . Among the \noindent243 hyphenelegant 288 period method of Karamata $ ( Wiener ˆ{ 18 } , $ p . 5 1 ) requires only one t extbook theorem : others\quad whothe have Weierstrass spread these approximation abstract ideas $ one\ l e shouldf t . theorem mention\ begin the{ a names l i g n e d of} &’’ I . \\ 2 0 sub AGelfand period Beurling , R . Godement comma Un and th acute-e S . L or . Segal e-grave . me sur les fonctions born acute-e es uniform e-acute ment continues&. sur\end l quoteright{ a l i g n e d axe}\ right . $ 19 R . Pitt , General Tauberian Theorems , Proc . London Math . Soc . , vol . 44 ( 1 938 ) 243 - r acute-e el commaH. Acta Math period comma 77 open parenthesis 1 945 closing parenthesis 1 27 hyphen 1 36 period The abstract288 . version of Wiener ’ s theory was given by $ Beurling ˆ{ 20 } . $ Among the others who20A. haveBeurling spread , Un th thesee´ or e` me abstract sur les fonctions ideas born onee´ es should uniform mentione´ ment continues the sur l ’ axe names ofr Ie´ .el Gelfand , Acta Math , . R , 77 . ( Godement 1 945 ) 1 27and - 1 36 S . . L . Segal .

$ 1 9 { H. }$ R . Pitt , General Tauberian Theorems , Proc . London Math . Soc . , vol . 44 ( 1 938 ) 243 − 288 .

\ hspace ∗{\ f i l l } $ 2 0 { A. }$ Beurling , Un th $ \acute{e} $ or $ \grave{e} $ me sur les fonctions born $ \acute{e} $ es uniform $ \acute{e} $ ment continues sur l ’ axe

\noindent r $ \acute{e} $ el , ActaMath . , 77 ( 1 945 ) 1 27 − 1 36 . 1 0 .... M period Tomi acute-c \noindentThe first one1 0 who\ h noticed f i l l M the . role Tomi of approximation $ \acute{c in} Tauberian$ the hyphen orems .. was .. Schmidt to the power of 8 comma .. when he .. studied .. the moment .. problem period .. Karamata \ hspaceintroduced∗{\ f the i l l approximation}The first using one who the Weierstrass noticed Theorem the role as the of main approximation in Tauberian the − t ool in the proof comma .. and in this way he added an important theorem t o the \noindentmathematicalorems literature\quad periodwas Wiener\quad quoteright$ Schmidt s approximation ˆ{ 8 } , of $ Fourier\quad transformswhen he was\quad studied \quad the moment \quad problem . \quad Karamata introducednot only a breakthrough the approximation in harmonic analysisusing the as a new Weierstrass method comma Theorem but also as a the main tsource ool in of new1 the 0 discoveries proof , period\quad .. Theand crucial in this role waythere he was added played by an approxima important hyphen theoremM . t Tomi o thec´ mathematicalt ions of various literature kindsThe in Tauberian first . one Wiener theorems who ’noticed speriod approximation the role of approximation of Fourier transforms in Tauberian was the - 8 notOn onlythe otherorems a breakthrough hand comma was theSchmidt inweakening harmonic, when of conditions analysis he studied for convergence as a newthe moment inmethod Taube hyphen , problembut also . a Kara- sourcerian theorems ofmata new i s discoveriesnot introduced only a formal the . matter\ approximationquad commaThe crucial .. but using raises role the the Weierstrass question there was of the Theorem played by as approxima the main − tscop ions e of of inverset various ool theoremsin the kinds proof in general in , Tauberian period and in .. Landau this theorems way replaced he . added Littlewood an important quoteright s theorem condi hyphen t o the t ion na submathematical n = O open parenthesis literature 1. Wienerclosing parenthesis ’ s approximation by the one of hyphen Fourier sided transforms condition :was na sub not n greater equal Onminus the K and other inonly this hand a way breakthrough , the weakening in harmonic of conditions analysis as for a new convergence method , but in also Taube a source− of rianhe made theorems annew important discoveries i s not step in only . Tauberian a The formal crucial theorems matter role period there , ..\ Accordingquad was playedbut to raises Wiener by approxima comma the question - t ions of of the scopthe most e of importantvarious inverse kinds .. conditions theorems in Tauberian for in convergence general theorems are . \ those .quad whichLandau .. originate replaced Littlewood ’ s condi − tfrom ion Schmidt $ na On to{ the then power} other= of 8 handO comma ( , the Schmidt 1 weakening )$ quoteright bytheone of s conditions conditions− sided comma for convergence conditioncontrary t o those in $ Taube : containing na - { onlyn }\ one geq − elementK$ comma andrian in theorems are this such thatway i s one not should only a consider formal a matter group of , elements but raises period the .. These question condi ofhyphen the scop het ions made can ane bof e important expressed inverse theorems in step the simplest in in Tauberian general way by using . theorems Landau slowly varying replaced . \quad sequences LittlewoodAccording comma ’ to s condi Wiener - t ,ion the most important \quad conditions for convergence are those which \quad o r i g i n a t e where thena sequencen = sO sub(1) nby = the a sub one 0 plus - sided a sub condition 1 plus period: periodnan period≥ plus −K aand sub nin comma this way such that he limint p right arrow infinitymade to the anpower important of open parenthesis step in Tauberian s p minus s subtheorems q closing . parenthesis According = 0 comma to Wiener where , the \noindentp right arrowmostfrom infinity important $ Schmidt comma plus ˆconditions{ q8 right} arrow, for $ convergence infinitySchmidt comma ’ s are q conditions greater those which equal , p contrarycomma originate and t w o slash those p right containing arrow 1 only one 8 commaelement i s s lowly ,from are varying suchSchmidt period that, Schmidt one should ’ s conditions consider , a contrary group of t o elements those containing . \quad onlyThese one condi − tSequence ions can ofelement this b e kind expressed , arewere such introduced inthat the one bySchmidt simplest should under consider way the by name a using group quotedblright slowly of elements .. varying seh . These sequences condi , wherelangsam the ..- oscillierende sequence t ions can FolgenRow b $ e s expressed{ n 1} quotedblright in= the a simplest{ Row0 } 2 period way+ by aFolgen using{ 1 .. slowly} they+ .. varying are . .. called . sequences ... slowly + , .. a varying{ n ..} se , $ such that $ \lim { p \rightarrow \ infty }ˆ{ ( s }(s p − s { q } ) = 0 , $ hyphen where the sequence sn = a0 + a1 + ... + an, such that limp→∞ p − sq) = 0, where wherequences periodp → ∞ .., Already+q → ∞ Landau, q ≥ p, toand thew/p power→ of1, 7i uses s s thislowly name varying quotedblright . .. seh .. langsam .. Folge n quotedblright .. O .. bvi Sequence of this kind were introduced by Schmidt under the name ” seh \noindentously comma$ functions p \rightarrow satisfying the condition\ infty 00 , + q \rightarrow \ infty , q \geq p langsam oscillierende F olgen T oday they are called slowly varying , $limint and right $ arrow w / x minus p y\ 0rightarrow comma x right. arrow1, infinity $ i to s the s lowly power of varying open square . bracket f open parenthesis y closing parenthesisse - minus f open parenthesis x closing parenthesis closing square bracket = 0 7 Sequenceare also called ofquences this slowly kind. varying Already were period introducedLandau .. Schmidtuses used by this Schmidt some name of the under” properties seh the commalangsam name which ’’ \quad Folgeseh n ” O langsamappear in\ calculationsquadbvi ouslyoscillierende ,comma functions usually satisfying $ in\ l studying e f t . Folgenthe conditions condition\ begin for{ convergencearray }{ c} of’’ the\\ . \end{ array }Today\ right . $ \quadgeneralthey kind\quad commaare which\quad showedc thea l l e importance d \quad ofslowly such functions\quad invarying Tauberian\quad se − [f(y) theorems period lim −f(x)] = 0 \noindent quences . \quad Already $→ Landaux 0,x→∞ ˆ{ 7 }$ uses this name ’’ \quad seh \quad langsam \quad Folge n ’ ’ \quad O \quad bvi In .. the .. proof .. of .. Littlewood quoteright s ..−y theorem .. 1928 hyphen 1929 comma .. Karamata .. noticed ously , functions satisfying the condition Schmidt quoterightare also s called slowlyslowly varying varying functions . commaSchmidt but he left used aside some convergence of the conditions properties , which in Tauberianappear theorems in calculations and started original , usually investigations in studying on theconditions fundamen for hyphen convergence of the gen- \ [ tal\lim structure{\rightarrow and meaning of these{ x functions{ − { periody }}} Nobody0 suspected , x that\rightarrow under \ infty }ˆ{ [ f ( y ) } −eralf kind ( , which x showed ) ] the = importance 0 \ ] of such functions in Tauberian theorems the surface. of this notion comma .. almost superficial after all comma lies a notion of real analysis commaIn .. not the .. clearly proof defined of though Littlewood it is in fact ’ s .. almost theorem .. elementary 1928 - period 1929 , .. The Karamata significance and the wide use of this notion comma so many years after its appear hyphen \noindent noticedare also Schmidt called ’ sslowly slowly varying functions . \quad ,Schmidt but he left used aside some convergence of the properties condi- , which ance commations and inso manyTauberian years aft theorems er the death and of started Karamata original comma investigations speaks about its on role the fundamen - appearin real analysis in calculations comma not only , usuallyin Tauberian in convergence studying conditions conditions comma for but convergence also in of the general kindtal structure , which and showed meaning the ofimportance these functions of such . Nobody functions suspected in Tauberian that under the theoremssurface . of this notion , almost superficial after all , lies a notion of real analysis , not clearly defined though it is in fact almost elementary . The In \quad thesignificance\quad proof and the\quad wideo use f \quad of thisLittlewood notion , so many ’ s \ yearsquad aftertheorem its appear\quad -1928 ance− 1929 , \quad Karamata \quad noti ced Schmidt ’, sand slowly so many varying years aftfunctions er the death , but of he Karamata left aside , speaks convergence about its roleconditions in real in Tauberiananalysis theorems , not only and in started Tauberian original convergence investigations conditions , on but the also fundamen in − tal structure and meaning of these functions . Nobody suspected that under the surface of this notion , \quad almost superficial after all , lies a notion of real a n a l y s i s , \quad not \quad clearly defined though it is in fact \quad almost \quad elementary . \quad The significance and the wide use of this notion , so many years after its appear − ance , and so many years aft er the death of Karamata , speaks about its role in real analysis , not only in Tauberian convergence conditions , but also in Jovan Karamata .... 1 1 \noindentthe final formJovan of many Karamata other theorems\ h f i l l period1 1 .... Karamata t ook from Schmidt .... open parenthesis or already from Landau to the power of 7 closing parenthesis .. the t erm itself for such functions semicolon .. he called them \noindentquotedblrightthe slowly final decreasing form of open many parenthesis other increasing theorems closing . \ parenthesish f i l l Karamata functions t open ook parenthesis from Schmidt functions\ acute-ah f i l l ( or croissance lente closing parenthesis quotedblright and \noindentdenoted themalready by L open from parenthesis $ Landau x closing ˆ{ 7 parenthesis} ) $ comma\quad ..the Schmidt t erm stated itself some properties for such of thesefunctions functions ; \quad he called them ’’open slowly parenthesis decreasing in the form ( of increasing sequences closing ) functions parenthesis ( which functions were almost $ evident\acute comma{a} $ but croissance Karamata was lente the ) ’’ and denotedthembyfirst to discoverJovan theKaramata structure $L ( of these x functions ) ,$ L open\quad parenthesisSchmidt x closing stated parenthesis some properties and their fundamental1 of 1 these functions (properties in thethe form period final of .. Schmidt form sequences of introduced many ) other which these theorems were functions almost .using Karamata differences evident comma , took but fromwhile Karamata Lan Schmidt hyphen was the ( or 7 firstdau had to alreadyalready discover expressed from theLandau them structure through) the theof t quotient these erm itself functions f open for parenthesis such functions $ L y closing ( ; x parenthesis he ) called $ andslash them theirf open ” parenthesis fundamental x closingproperties parenthesisslowly . \ periodquad decreasing FinallySchmidt comma ( increasing introduced ) functions these functions ( functions usinga´ croissance differences lente , ) ”while and Lan − dauhadKaramata alreadydenoted open square expressed them bracket by 1L them( 5x) closing, throughSchmidt square bracketstated the quotient commasome properties open $ square f (of bracket these y 37 functions ) closing / square f ( in ( bracket x gave ) a really. $ precise F i n a lthedefinition l y ,form : of .. asequences function L ) open which parenthesis were almost x closing evident parenthesis , but .. Karamata defined was the first Karamatafor x greaterto [ 1equal discover 5 ] 0 is , s the [ lowly 37 structure varying] gave if of a it these reallyis positive functions precise commaL ..(x definition continuous) and their such fundamental : that\quad afunction properties $L ( x ) $Equation:\quad. opend e f i Schmidt parenthesisn e d introduced 1 closing parenthesis these functions .. L sub using L sub differencesopen parenthesis , while x closing Lan parenthesis- dau had to the power of openf o r parenthesis $ x already\ lambdageq expressed x0 closing $ is parenthesis them s lowly through right varying arrowthe quotient 1 if comma it isf x( righty positive)/f( arrowx). Finally infinity , \quad , Karamatacontinuous [ 1 5 such that for every fixed] , [ lambda37 ] gave greater a really equal precise 0 period definition : a function L(x) defined for x ≥ 0 \ beginIt i s{ Karamataa l i gis n s∗} lowly quoteright varying s big if achievement it is positive that , he derived continuous a whole such theory that from L this{ simpleL }ˆ{ definition( \lambda open squarex bracket ) } 1{ 5 closing( x square ) }\ bracketrightarrow comma open square1 bracket , x 37\ closingrightarrow square bracket\ infty \ tag ∗{$ ( 1 ) $} period F-r sub om open parenthesis 1 closing parenthesis(λx) Karamata obtained the canonical \endrepresentation{ a l i g n ∗} for such functions : LL (x) → 1, x → ∞ (1) Equation: open parenthesis 2 closing parenthesis .. L open parenthesis x closing parenthesis = c open parenthesis x closing \noindent forfor every every fixed fixedλ ≥ $0. \lambda \geq 0 . $ parenthesis exponentIt i s integral Karamata sub 0 ’ to s big the achievementpower of x epsilon that open he derived parenthesis a whole u u closing theory parenthesis from this du sim- comma where c open parenthesis x closing parenthesis right arrow c greater 0 and epsilon open parenthesis x closing parenthesis It i s Karamataple definition ’ s big[15] achievement, [37]. F − rom(1) thatKaramata he derived obtained a whole the canonical theory from representation right arrow 0for as x such right functions arrow infinity : comma and also another fundamental thissimpledefinitionproperty open parenthesis about $[ uniform 1 convergence 5 ] closing , [ parenthesis 37 ] : the . relation F−r open{ parenthesisom } ( 1 closing1 ) paren- $ Karamata obtained the canonical thesisrepresentation holds uniformly for with such functions : resp ect t o lambda in open square bracket a comma b closingZ x square bracket for fixed 0 less a less or equal b less infinity \ begin { a l i g n ∗} L(x) = c(x) exp ε(uu)du, (2) period Some of the properties of slowly 0 Lvarying ( functions x ) can = be used c as ( the x initial ) definition\exp period\ int ˆ{ x } { 0 }\ varepsilon ( { u } u ) duThe , creation\ tagwhere∗{ of$ the (c( theoryx) 2→ c of > ) slowly0 $and} varyingε(x) → functions0 as x → i s ∞ a, landmarkand also period another fundamental property \endThere{ a l i are g n ( not∗} about many uniform theories t convergence oday which are ) : so the general relation comma ( 1 .. ) and holds which uniformly arise with resp ect t o from suchλ a∈ simple[a, b] definitionfor fixed as0 open< a ≤ parenthesisb < ∞. Some 1 closing of the parenthesis properties period of slowly varying functions \noindentAs stated beforecanwhere be comma used $ c aas simple the( initial x proof ) can definition sometimes\rightarrow . give newc results> period0 $ .. Kara and hyphen $ \ varepsilon ( x ) \rightarrowmata replaces conditions0 $The as of creation $ the x form\rightarrow ofx to the the theory power of\ infty kslowly lg sub varying k, to $ the andfunctions power also of alpha i another s a x landmark comma fundamental where . the it erated logarithmpropertyThere ( about are uniform not many convergence theories t oday ) : which the relationare so general ( 1 , ) holds and which uniformly arise from with respappears e c t insuch t Hardy o a $ hyphen simple\lambda Littlewood definition\ in theorem as ([ 1 ) open a . parenthesis , b ]$1 9 1 3 forfixed closing parenthesis $0 comma< bya L open\ leq parenthesisb < x\ infty closing parenthesis. $ SomeAs .. stated and of in the this before way properties , he a simple of proof slowly can sometimes give new results . Kara - varying functions can be used as the initialk α definition . gives hyphenmata this replaces is the opinion conditions of many of the authors form hyphenx lgk anx, where important the theorem it erated open logarithm square bracket appears 19 closing square bracket : in Hardy - Littlewood theorem ( 1 9 1 3 ) , by L(x) and in this way he gives - \ hspaceLet U open∗{\thisf parenthesis i l l is}The the opinioncreation x closing ofparenthesis ofmany the authors .. theory be a nondecreasing - anof important slowly function varying theorem in open functions [ square19 ] : bracket i s 0 a comma landmark infinity . closing parenthesis such that Let U(x) be a nondecreasing function in [0, ∞) such that \noindentw open parenthesisThere are x closing not parenthesis many theories = integral t oday sub 0 to which the power are of so infinity general e to the , \ powerquad ofand minus which xs d open arise brace from such a simple definition as ( 1 ) . Z ∞ U open parenthesis s closing parenthesis closing bracew(x) = e−xsd{U(s)} is bounded for every x greater 0 and if rho greater equal0 0 comma L open parenthesis x closing parenthesis .. is slowly \ hspace ∗{\ f i l l }As stated before , a simple proof can sometimes give new results . \quad Kara − varying and wis open bounded parenthesis for every x closingx parenthesis > 0 and if = ρ ≥ 0,L(x) is slowly varying and w(x) = x to the powerx−ρL(1 of/x minus) as rhox L→ open0+, parenthesisthen U(x 1) slash∼ xρL x(x closing)/Γ(ρ + parenthesis 1), x → ∞. ..Analogously as x right arrow , 0 plus comma then U open\noindent parenthesismata x closing replaces parenthesis conditions thicksim xof to the the power form of $ rho x L ˆ{ openk }\ parenthesislg ˆ{\ x closingalpha parenthesis} { k } slashx Capital , $ Gammawhere theopen itparenthesis erated rho logarithm plus 1 closing parenthesis comma x right arrow infinity period Analogously comma appears in Hardy − Littlewood theorem (1913) ,by $L ( x )$ \quad and in this way he g i v e s − this is the opinion of many authors − an important theorem [ 19 ] :

\ centerline { Let $ U ( x ) $ \quad be a nondecreasing function in $ [ 0 , \ infty ) $ such that }

\ [ w ( x ) = \ int ˆ{\ infty } { 0 } e ˆ{ − xs } d \{ U ( s ) \}\ ]

\noindent is bounded for every $ x > 0 $ and i f $ \rho \geq 0 , L ( x ) $ \quad is slowly varyingand $w ( x ) =$ $ x ˆ{ − \rho } L ( 1 / x ) $ \quad as $ x \rightarrow 0 + , $ then $ U ( x ) \sim x ˆ{\rho } L ( x ) / \Gamma ( \rho + 1 ) , x \rightarrow \ infty . $ Analogously , 1 2 .... M period Tomi acute-c \noindentL open parenthesis1 2 \ h f 1 i lslash l M x . closing Tomi parenthesis $ \acute ..{ canc} $ be rep laced by L open parenthesis x closing parenthesis .. and x right arrow 0 by x right arrow infinity period \noindentA function$ of Lthe form ( R 1 open / parenthesis x ) $x closing\quad parenthesiscanbereplacedby = x to the power of $L rho L ( open x parenthesis )$ \ xquad closingand parenthesis$ x \rightarrow i s then called regularly0 $ by varying $ x \rightarrow \ infty . $ with the regularity index rho period It s definition i s given comma analogously to open parenthesis 1 closing parenthesis comma\ hspace by∗{\ f i l l }Afunctionoftheform $R ( x ) = xˆ{\rho } L ( x ) $ i s then called regularly varying Equation:1 open 2 parenthesis 3 closing parenthesis .. limint lambda right arrow infinity R subM R . Tomi sub openc´ parenthesis lambda\noindent closingLwith(1 parenthesis/x) thecan regularity to be the rep power laced of index by openL parenthesis( $x) \rhoand lambdax.→ $0 xby It closing sx definition→ parenthesis ∞. = i x sto given the power , of analogously rho period to ( 1 ) , by ρ for some rho from theA interval function .. minus of the infinity form lessR( rhox) less = x infinityL(x) i period s then .. called If rho liregularly es between varying two finite \ beginbounds{ a thenl i gwith n such∗} the a function regularity i s called index a regularlyρ. It s definition bounded open i s parenthesis given , analogously R minus 0 closing to ( 1 parenthesis ) , by function period \limThe properties{\lambda of such\ functionsrightarrow .. open\ parenthesisinfty } RR minus{ R 0} closingˆ{ ( parenthesis\lambda commax .. ) as} well{ as( their\lambda definition ) } = x ˆ{\rho } . \ tag ∗{$ ( 3 ) $} comma .. was (λx) ρ \end{ a l i g n ∗} lim RR (λ) = x . (3) first given by Avakumovi acute-c to the power ofλ→∞ 2 1 sub comma .. a student of Karamata period u-F sub nctions from the class open parenthesis R minus 0 closing parenthesis can also be defined through inequalities : .. if there exist number \noindentgamma greaterforf o r some 0 some and Capitalρ from $ \rho Gamma the$ interval fromgreater the 0 such−∞ interval that< x to ρ the\ 0 and as given KaramataΓ > 0 bysuch quoteright Avakumovi that xγ sg Tauberian(x) $i s\acute almost Theorem{c increasing} ˆ period{ 2 .. and 1 The} x{ proof−Γg, (x} of)$almost Littlewiid\quad decreasing quoterighta student s of Karamata $ .Theorem u−F comma,{ thenn c t..g i o( thex n) sb introduction} elongs$ from t o of the the the class slowly( varyingR − 0). functions L open parenthesis x closing parenthesis comma .. i periodc l a s se period $ ( ..Karamata the R − ’0 s theorem ) $ can , mentioned also be above defined , with through a regularly inequalities varying function : \quad i s if there exist number creation ofknown the theory today of slowly as Karamata varying functions ’ s Tauberian comma and Theorem Karamata . quoteright The proof s Tauberian of Littlewiid ’ s \noindenttheorem areTheorem$ three\gamma results , of the> lasting introduction0 $ value and in the $of mathematical\ theGamma slowly> varying lit erature0$ functions commasuch thatL(x), $xˆi .{\ e .gamma the } g ( xwhich ) $ cannot icreation s be almost left of out the increasing period theory .. Karamata of and slowly quoteright $ xvarying ˆ{ − s Tauberian functions \Gamma theorem ,} andg Karamata i s ( the first x comma ’ )$ s Tauberian and almost decreasingalso the nicesttheorem , example then are of$g three an application results ( x of of )$ lasting slowly belongs varying value functions in tothethe mathematical period class Instead $( lit erature R − , which0 ) . $ of somethingcannot limited be hyphen left out the . it erated Karamata logarithm ’ s hyphen Tauberian in this theorem problem i of s Hardy the first and , and also the KaramataLittlewood ’nicest .. s we theorem ..example have .. , theof mentioned an function application .. above R open of , slowly parenthesis with varying a regularly x closing functions parenthesis varying . Instead = function x to of the something power of gamma L open parenthesisi s known xlimited closing today parenthesis - as the Karamata it erated .. whichlogarithm ’ i s s .. Tauberian as .. -general in this Theorem .. problemas . \ ofquad HardyThe and proof Littlewood of Littlewiid we ’ s Theorempossible period ,have\quad thethe function introductionR(x) of = thexγ L slowly(x) which varying i s functions as general $ L as ( possible x ) , $ \quad i . e . \quad the creationAmong the. of three the results theory mentioned of slowly above comma varying .. there functions is no doubt , thatand the Karamata ’ s Tauberian theoremmost important areAmong three place is resultsthe occupied three ofby results the lasting theory mentioned of value slowly above in varying the , functions mathematical there is period no doubt lit that erature the most , whichTheories cannot haveimportant their be own left place destiny out is occupied. comma\quad justKaramata by like the bookstheory period’ s of Tauberian ..slowly Karamata varying theorem published functions ithe s the . Theories first , and alsofirst article thehave nicest on slowly their example varying own destiny functions of an , applicationjust in an like almost books unknown of . slowly j Karamata ournal varying open published parenthesis functions the Math first . hyphen Instead article ofematica something commaon slowly limited Cluj closing varying− parenthesisthe functions it erated period in an .. logarithm The almost revised unknown− andin enlarged this j ournal version problem ( open Math of square Hardy - ematica bracket and 37 , closing square bracketLittlewood of thisCluj paper\quad ) .hadwe no The\quad revisedhave and\quad enlargedthe version function [ 37\quad ] of this$ paper R ( had x no particular ) = x ˆ{\gamma } Lparticular ( xresp resp ) onse $onse\ eitherquad either periodwhich . .. For For i s almost almost\quad 25 25as years years\quad both both articlesg e n articles e r a remained l \quad remained obas hyphen ob - scure , and p oscure s s i b comma l ewere . and almost were almost forgotten forgotten . period Neither .. Neither Karamata Karamata nor nor his his students students comma , who who started in started inthe fifties fifties t o t investigate o investigate these these functions functions comma ,used used the the t erm t erm .. quotedblright ” theor .. theor of slowly Amongof slowly the varyingvarying three functions results functions quotedblright mentioned ” . Foreign period above authors .. Foreign , \quad gave authorsthere an gave appropriate is an no appropriate doubt name that name to this theto theory mostthis theory important, comma and emphasized place and emphasized is occupied it s importanceit s importance by the . period theory of slowly varying functions . TheoriesIn .. 1966 havethe large theirIn monograph own1966 destiny the of Feller large to , monograph the just power like of 22of books ..Feller on probability22 . \quadon probability theoryKaramata ant it theory published s ant it the s first article on slowly varying functions in an almost unknown j ournal ( Math − 2 1 sub V period G period21V. AvakumoviG . Avakumovi acute-cc´, On sub the comma behavior On of the Laplace behavior integrals of Laplace on the integralsboundary on of converthe boundary - of conver hyphenematica ,gence Cluj ( Ph ) . . D\ .quad ThesisThe , in Serbian revised ) , Beograd and enlarged , 1 940 ; Math version . Zeit . 46 [ ( 37 1 940 ] )of , 62 this - 66 , paper 67 - 69 ; had no particulargence opensee parenthesis resp onse Ph either period D . period\quad ThesisFor comma almost in 25 Serbian years closing both parenthesis articles comma remained Beograd ob comma− 1 940 scure , and were almost forgotten . \quad Neither Karamata nor his students , who semicolon MathKaramata period [Zeit 5 1 period] : Bemerkung 46 openu´ parenthesisber die vorstehende 1 940 closing Arbeit parenthesis des Herrn Avakumovi comma 62c´ hyphen. 66 comma 67 hyphen 69 started in the fifties t o investigate these functions , used the t erm \quad ’’ \quad theor semicolon see 22W. Feller , An Introduction to Probability Theory and its Applications , I , II , Wiley and Sons , ofKaramata slowlyNew open varying York square , 1 functions 971 bracket . 5 1 closing’’ . \quad squareForeign bracket : Bemerkung authors gave u-acute an ber appropriate die vorstehende name Arbeit to des Herrn Avakumovithis theory c-acute , sub and period emphasized it s importance . 2 2 sub W period Feller comma An Introduction to Probability Theory and its Applications comma I comma II comma Wiley\ hspace ∗{\ f i l l } In \quad 1966 the large monograph of $ Feller ˆ{ 22 }$ \quad on probability theory ant it s and Sons comma New York comma 1 971 period \ hspace ∗{\ f i l l } $ 2 1 { V. }$ G . Avakumovi $ \acute{c} { , }$ On the behavior of Laplace integrals on the boundary of conver −

\noindent gence ( Ph . D . Thesis , in Serbian ) , Beograd , 1 940 ; Math . Zeit . 46 ( 1 940 ) , 62 − 66 , 67 − 69 ; see

\noindent Karamata [ 5 1 ] : Bemerkung $ \acute{u} $ ber die vorstehende Arbeit des Herrn Avakumovi $ \acute{c} { . }$

$ 2 2 { W. }$ Feller , An Introduction to Probability Theory and its Applications , I , II , Wiley and Sons , New York , 1 971 . Jovan Karamata .... 1 3 \noindentapplicationsJovan came outKaramata period ..\ Fellerh f i l l introduced1 3 Karamata quoteright s ideas and theorems about slowly varying functions in this field semicolon .. he introduced the means and \noindentt ools usedapplications t o present more came clearly out many . theorems\quad Feller and t o give introduced them their fiKaramata hyphen ’ s ideas and theorems aboutnal form slowly period varying This was the functions source of new in results this period field Karamata ; \quad washe never introduced interested the means and tin ools probability used theory t o present comma not more even clearly in the basic many facts theorems of this branch and period t o .. give But many them their $ fi − $ nalmathematicians form . This who was worked the in source probability of theory new results have seen at . onceKaramata that was never interested the notionJovan of regularly Karamata varying function and its properties are well adapted 1 3 \noindentt o many problemsapplicationsin probability in this came area theory period out . .. ,With not Feller this even notion introduced in the the final basic Karamata statements facts ’of s the of ideas this and branch theorems . \quad But many mathematiciansprincipal theoremsabout slowly who then worked get varying a simple in functions form probability period in this .. This theory field is the ; case have he introducedfor seen example at with theonce means that and t ools thethe notion Centralused Limit of t regularly Theoremo present of more Khinchine varying clearly comma function many Feller theorems and comma its L and acute-e properties t o sub give vy them comma are their well 1 935fi adapted− commanal form where neces hyphen tsary o many and sufficient. problems This was conditions in the this source are nowarea of expressed new . \quad results in theWith . languageKaramata this of notion regular was never the interestedfinal statements of the principalvariation periodin theorems probability .. This theni salso theory get the case , a not simple of even the Weak form in the Law . basic of\quad Large factsThis Numbers of isthis comma the branch case where . for But example many with thenecessary Central andmathematicians sufficientLimit Theorem conditions who of for worked Khinchine the convergence in probability , Feller of arithmetic theory , L means $ have\acute seen{e} at{ oncevy that , } the1 935 , $ whereS sub neces n slashnotion− n right of arrow regularly mu of varying random functionvariables x and sub itsi are properties given comma are S wellsub n adapted being the sumt o many of these ran hyphen sarydom andvariablesproblems sufficient period Forin this conditionscritical area branching . are With processes now this expressed necessary notion the and in final sufficient the statements language con hyphen of the regular principal v aditions r i a t i ofor ntheorems iterated . \quad solution thenThis of get ithe s afunctional also simple the equation form case . f ofopen the This parenthesis Weak is the Law x closingcase of for Large parenthesis example Numbers = xwith minus , where the a open parenthesis necessary and sufficient conditions for the convergence of arithmetic means x closing parenthesisCentral R Limitopen parenthesis Theorem x of closing Khinchine parenthesis , Feller , L e´vy, 1935, where neces - sary $are S given{ n commaand} sufficient/ where n a open\ conditionsrightarrow parenthesis are xnow\ closingmu expressed$ parenthesis of random in the right variables language arrow a comma of regular$ x and{ Ri variation open}$ parenthesis are . given x closing $ , parenthesisS { n }$ i sThis a being regularly i s thealso varying sum the case offunction these of the of the ran Weak− Law of Large Numbers , where necessary and dom variables . For critical branching processes necessary and sufficient con − form x to thesufficient power of conditions k L open parenthesis for the convergence x closing parenthesis of arithmetic period Then means fromS xn/n sub→ nµ plusof 1 random = f open parenthesis x ditions for iterated solution of the functional equation $ f ( x ) = x − a sub n closingvariables parenthesisx iti followsare given that,S x subn being n thicksim the sum a sub of k these n to the ran power - dom of minus variables k sub . 1 For L open critical parenthesis 1 slash n( closing x parenthesis )branching R comma ( processes x ) $ necessary and sufficient con - ditions for iterated solution of n right arrowthe infinity functional period equation Karamataf ....(x) open = x square− a(x)R bracket(x) 1 4 closing square bracket .... uses this functional equation as\noindent an examplearearegiven and given , where ,wherea(x) $a→ a, and (R( xx) i s) a regularly\rightarrow varyinga function ,$and$R of the ( x )$ i s a regularlyk varying function of the −k1 considers itform as a mathematicalx L(x). Then problem from x ofn+1 lesser= f importance(xn) it follows comma that whilexn Seneta∼ akn to theL(1/n power), of 23 uses this solutionn → ∞ as. aKaramata basic fact in some [ 1 probabilistic4 ] uses problems this functional period .. equation Such kind as an example and 23 \noindentof a completeconsidersform accordance $ x it ˆ as{ of ak the mathematical} problemL ( with x the problem solution ) .$ of can lesser be Thenfrom found importance in $x , while{ nSeneta + 1 uses} = f ( x many{ n places} this)in $ thesolution it literature follows as awhere basic that regularly fact $ x in varying{ somen }\ functions probabilisticsim area involved problems{ k } periodn . ˆ{ − Suchk kind{ 1 of}} a L ( 1 /This n can ) becomplete seen , $ from accordance many cit ations of the of Karamata problem commawith the esp solution ecially of can the befunction found in many places denoted byin L the open literature parenthesis where x closing regularly parenthesis varying period functions are involved . This can be seen \noindentForty .. sixfrom$ years n many ..\ afterrightarrow cit the ations first .. of work Karamata\ infty .. of Karamata ,. esp $ commaecially Karamata .. of the the\ monographh function f i l l [ 1 by 4 ] \ h f i l l uses this functional equation as an example and Seneta to the power of 23 about regularly varying functions appeared period .. Seneta gave a concise \noindentand .. substantialconsiders .. survey it .. as of a all mathematical results .. connected problem .. with Karamata of lesser functions importance , while $ Seneta ˆ{ 23 }$ usesknown this up t solutiono that t ime as comma a basic as well fact as their in historical somedenotedby probabilistic developmentL(x). period problems .. But he also. \quad Such kind ofintroduced a complete numerous accordance applications of : .. the from problem number theory with through the solution integrals can be found in many places inForty the literature six years afterwhere the regularly first work varying of Karamata functions , are the involved monograph . with parametersby comma which reduce to the calculations with these functions comma t o ThisTauberian can be theorems seen and from probability many cit theory ations period of .. There Karamata were numerous , esp ecially exam hyphen of the function Seneta23 about regularly varying functions appeared . Seneta gave a concise and ples and also some generalizations of the notion of these functions period .. Already \ begin { a l i gsubstantial n ∗} survey of all results connected with Karamata functions in .... 1 949known comma up .... t Korevaar o that t comma ime , as.... well van .... as Aardenne their historical hyphen Ehrenfest development .... and . .... But de .... he Brujin also to the power of 24denoted comma .... proved by ....L the ( x ) . \end{ a l i g n introduced∗} numerous applications : from number theory through integrals with 2 3 sub Eparameters period Seneta , comma which Regularly reduce to Varying the calculations functions comma with Lecture these functions Notes in Mathematica , t o comma 5 80 comma Springer comma \ hspace ∗{\Tauberianf i l l } Forty theorems\quad s and i x yearsprobability\quad theoryafter . the There first were\quad numerouswork exam\quad -of ples Karamata , \quad the monograph by Berlin commaand Heidelberg also some comma generalizations New hyphen of York the commanotion 1 of 9 these 76 semicolon functions Karamata . Already quoteright s iteration theorems and normed regularly in 1 949 , Korevaar , van Aardenne - Ehrenfest and de Brujin24, proved the \noindentvarying sequences$ Seneta in a historical ˆ{ 23 } light$ commaabout Publ regularly period Inst varying period Math functions period Beograd appeared 48 open . \ parenthesisquad Seneta 62 closing gave a concise 23 Seneta , Regularly Varying functions , Lecture Notes in Mathematica , 5 80 , Springer , Berlin parenthesisand \quad opensubstantial parenthesisE. 1 990\quad closingsurvey parenthesis\quad 45 hyphenof all 5 results1 period \quad connected \quad with Karamata functions , Heidelberg , New - York , 1 9 76 ; Karamata ’ s iteration theorems and normed regularly varying known24 sub up van t Aardenne o that hyphen t ime Ehrenfest , as well T period as their comma historicalde Bruj in N period development G period comma . \quad KorecaarBut he J period a l s o comma sequences in a historical light , Publ . Inst . Math . Beograd 48 ( 62 ) ( 1 990 ) 45 - 5 1 . A note on slowly oscillating 24 Aardenne - Ehrenfest T . , de Bruj in N . G . , Korecaar J . , A note on slowly oscillating \noindentfunctions commaintroduced Nieuwvan period numerous Arch period applications Wisk period : 23\quad open parenthesisfrom number 1 949 theory closing parenthesis through 77 integrals hyphen 86 with parametersfunctions , , Nieuw which . Arch reduce . Wisk to. 23 the ( 1 949 calculations ) 77 - 86 with these functions , t o

\noindent Tauberian theorems and probability theory . \quad There were numerous exam − ples and also some generalizations of the notion of these functions . \quad Already

\noindent in \ h f i l l 1 949 , \ h f i l l Korevaar , \ h f i l l van \ h f i l l Aardenne − Ehrenfest \ h f i l l and \ h f i l l de \ h f i l l $ Brujin ˆ{ 24 } , $ \ h f i l l proved \ h f i l l the

$ 2 3 { E. }$ Seneta , Regularly Varying functions , Lecture Notes in Mathematica , 5 80 , Springer , Berlin , Heidelberg , New − York , 1 9 76 ; Karamata ’ s iteration theorems and normed regularly varying sequences in a historical light , Publ . Inst . Math . Beograd 48 ( 62 ) ( 1 990 ) 45 − 5 1 .

\ hspace ∗{\ f i l l } $ 24 { van }$ Aardenne − Ehrenfest T . , de Bruj in N . G . , Korecaar J . , A note on slowly oscillating

\noindent functions , Nieuw . Arch . Wisk . 23 ( 1 949 ) 77 − 86 14 .... M period Tomi acute-c \noindentmain properties14 \ h of f i R l l hyphenM . Tomi functions $ under\acute the{c assumption} $ that L open parenthesis x closing parenthesis is Lebesque hyphen \noindentintegrablemain period properties Karamata and of his students $ R gave− $ new functions generalizations under comma the as assumption well as a that $ L ( x ) $multidimensional is Lebesque variant− for regularly varying functions period Twenty years aft er integrablethe death of Karamata . Karamata the monograph and his students of Bingham gave comma new .. Goldie generalizations and Teugels to the , aspower well of 25 as a multidimensionalwas published comma variant .. where this for theory regularly i s presented varying in detail functions and with supple . Twenty hyphen years aft er ments period14 .. This book comma published in the series Encyclopedia of mathematics and M . Tomi c´ \noindentits Applicationsmainthe death comma properties of.. contains Karamata of R all− resultsfunctions the upmonograph to under the t ime the of of assumption Bingham it s publication , \ thatquad periodLGoldie(x ..) Inis Lebesque and $ Teugels - ˆ{ 25 }$ wasincludes published 70integrable years ,of\ investigationquad . Karamatawhere in this this and area theory his comma students i and s these presented gave investigations new generalizations in detail show and , with as well supple as a − mentsthat the . t\quadmultidimensional ime wasThis not lost book period variant, published .. The for extensive regularly in litthe erature varying series about functions Encyclopedia slowly varying . Twenty of yearsmathematics aft er and 25 itsfunctions Applications speaksthe death it self , of about\quad Karamata itscontains numerous the monograph applications all results in of many Bingham up areas to the of , real t Goldie ime of and it sTeugels publicationwas . \quad In includesanalysis periodpublished 70 years .. Karamata , of investigation where is the most this frequently theory in i this scited presentedYugoslav area , in andmathematician detail these and investigations with period supple - ments show thatAt the the beginning. t ime This of was the book not20 th lost century , published . several\quad in mathematicalThe theextensive series theories Encyclopedia lit had erature of mathematics about slowly and varying its functionsinfluence onApplications speaks the future it development self , contains about of mathematics its all resultsnumerous period up applications to .. 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Riesz ,P . Montel , W.and Blachke conscientiousscientific , . comments . meetings . His period reviews and in conferences the Zentralblatt between f $ the\ddot{ twou} $ world r Mathematik wars , and in the JahrbuchAfter the ..he $ important met\acute many{ ..u works} $ scientists .. b from er die.. with 1930 Fortschritte whom hyphen he 1932 not comma der only Mathematik .. corresponded in .. the .. period were , .. noted 1932but alsohyphen for had their 1939 comma criticism Karamatasome had lit sort tle interest of friendly not only relationship for regularly - varying among functions others severalthem hyphen famous mathematicians \noindentselves commaofand that but conscientious also t ime for their , such applications ascomments : Landau period . , Knopp , I . Schur , Fej e´ r , M . Riesz , P . Let us rememberMontel that , W the . t Blachke ime when , Karamata . . . 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Tomi period $ \acute{c} $ Wiener t ook active part in the English School of Analysis at the time \noindentof it s heightsand comma summability with Hardy in and its Littlewood final formand with . the group of their most well hyphen known .. students period .. Everyday .. exchange .. of thoughts .. and ideas .. in their Wienercircle brought t ook newactive results part and methodsin the period English .. On School the other of hand Analysis comma .. at Karamata the time quoteright s ofmilieu it s was heights indifferent , .. with and not Hardy .. interested and Littlewood even in more and important with .. the things group comma of their most wleast e l l − ofknown all1 in 6 the\quad sciencestudents period .. In . every\quad milieuEveryday comma where\quad thereexchange i s no interest\quad commao f thoughts thereM . 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ThatBesides Karamata was the that time studied , working onof somebody feverishWiener work else : .. ’ t s ’ o new publishs ideas theory as soon i with s as not possiblethe the enthusiasm newsame results thing that with as was modern one typical ’ s of own him discovery - he gave . new In the pt e ools r i o d comma 1933scientific ..− and39 t contributions o , master he published old problems , used a period and large enlarged .. number During by these of the treatises few others years he , but published . \ hequad didnThat ’ t was reach the time ofmore feverish thanthe 20 articles work summit : semicolon\ ofquad scientific ont o the publish creativity other hand as comma of soon his youthas as Karamata possible ( 1 928 himself new - 1 said932 results comma ) . with he The was modern basic twriting o o l s the, ideas\quad paper and onand slowly discoveries t o varying master b functions elong old problems t for o Norberta years . comma\ Wienerquad obsessedDuring . with Besides these only that few , years working he on published moreone idea than andsomebody 20 with articles one thoughtelse ’ ; s on periodideas the i.. s otherAt not this the phand eriod same , he thing as was Karamata a as young one and ’ shimself unknownown discovery said , . he In wasthe p writingmathematician theeriod paper and 1933 had on - no39 slowly feeling , he published of varying t ime period a large functions It was number a period for of treatises whena years passion . , forobsessed That was with the time only of onetheidea unknownfeverish and was with hunting work one him: thought comma t o publish and . \quad when as soon heAt gave thisas the possible p most eriod of new himself he results was period awith young.. 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Karamata open square . applied bracket And just 55 closing square bracketthe notion periodas .. of H This . normal method R . families has Pitt shown , the in\quad 1938( Montel , reduced ) \quad Wiener[ 55 ’ s ] theory . \quad t oThis the simplest method has shown the relationshiprelationshipform b etween b , Karamata etween different different appliedcomma apparently his , new apparently method independent tindependent o theorems various comma theorems theorems but their of Tauberian , but their type corecore b ecame ecame. visible The visible .. range open\quad ofsquare this[ bracket theory 39 ] 39 . and closing\quad it s square importanceThe \ bracketquad s period were y n t h visible e .. s iThe s \ ..quad . synthesis Newof these ..elements of these works works\quad .. cancan .. \quad be \quad found in be .. found inappeared in this theory as Karamata applied the notion of normal families ( \noindenthis small monographMontelhis small ) published monograph[ 55 ] . by Hermann This published method .. open bysquarehas Hermann shown bracket the 60\quad closing relationship[ square 60 ] bracket b . etween\quad period differentBy .. applying By applying , Schmidt Schmidt ’ s quoterighttheorem s onapparently power series independent t o Stieltj theorems es ,but integrals their core and b to ecame the visible Laplace [ transform39 ] . The , hetheorem weakened onsynthesis power the series conditions oft o these Stieltj works esfor integrals convergence can and to be the in Laplace found Tauberian transform in 0 comma− theorem . \quad New resultshe weakened followedhis the small conditions from monograph for this convergence . published\quad inThis Tauberian by Hermann shows 0 hyphen the role [ theorem 60 ] of . period the By Stieltj .. applying New es Schmidt integral as an results followed’ s theorem from this on period power .. This series shows t o the Stieltj role of es the integrals Stieltj es andintegral to as the an Laplace transform , he weakened the conditions for convergence in Tauberian 0 - theorem . New results followed from this . This shows the role of the Stieltj es integral as an Jovan Karamata .. 1 7 \noindentanalytic t oolJovan period Karamata .. As this\ tquad ime Karamata1 7 was interested in Borel summability open parenthesis B hyphen summability closing parenthesis comma .. esp ecially in the inverse of Hardy hyphen Littlewood quoteright\noindent s theoremanalytic t ool . \quad As this t ime Karamata was interested in Borel summability (Bconcerned− summability with B hyphen ) , summability\quad esp period ecially .. The in lack the of a simpleinverse proof of of Hardy this theorem− Littlewood ’ s theorem concernedand the .. attempts with B ..− ofsummability many authors .. . T\ periodquad The .. Vijayaraghavan lack of a comma simple .. O proof period of .. Szasz this comma theorem .. and in andparticular the \quad of Hardyattempts and Littlewood\quad commaof many can be authors seen in Karamata\quad T. quoteright\quad sVijayaraghavan work in the , \quad O. \quad Szasz , \quad and in particularso called shiftingJovan of Karamata Hardy of indices and period1 Littlewood 7 .. Among numerous , can be problem seen in in the Karamata articles of that ’ s work in the sot ime called one cananalytic shifting find a t theorem ool of . indices of Asthat this typ . et\ commaquad ime KaramataAmong but with numerous new was ideas interested problemand beyond in inBorel the summability articles of ( that treal ime analysis oneB can comma- summability find in the a theorem area ) ,of complex esp of ecially that functions typ in the period e , inverse but .. Karamata with of Hardy new open ideas- square Littlewood and bracket beyond ’ 66 s theorem closing square bracket .. hasreal proved analysisconcerned , in with the B area - summability of complex . functions The lack . of\quad a simpleKaramata proof of [ 66this ] theorem\quad has proved thethe followingand theorem the theorem : attempts : of many authors T . Vijayaraghavan , O . Szasz , Let and in particular of Hardy and Littlewood , can be seen in Karamata ’ s work in \ centerlinef open parenthesisthe{ Let so} z called closing shifting parenthesis of indices= sum a . sub k Among z to the power numerous of k .. problem be convergent in the for articles .. bar z bar of less or equal 1 open parenthesisthat 4 closing t ime parenthesis one can find a theorem of that typ e , but with new ideas and beyond \ centerlineand let f openreal{ $ parenthesis analysis f ( , z in closingz the ) area parenthesis = of complex\sum .. be boundeda functions{ k in} K . :z bar ˆ Karamata{ z minusk }$ 1 slash\ [quad 66 2 ] barbe has = convergent 1 proved slash 2 period for In\ orderquad that$ \mid the serieszthe open\ followingmid parenthesis\ leq theorem 4 closing1 : parenthesis ( 4 ) $ } be B hyphen summable to s at z = 1 comma .. it is necessaryLet and sufficient that the Fourier \noindent andlet $f ( zP )$k \quad be bounded in $K : \mid z − 1 / s eries of f on K converges to sf at(z) the = sameakz pointbe period convergent for | z |≤ 1 (4) 2 Karamata\mid and conj= letectured 1f( /z the) general 2be bounded .$ problem Inorderthatthe in byK replacing:| z − 1/ K2 | with= 1/ a2 series. parabolaIn order (4) that the series ( 4 ) be behaving B − thesummabletoB imaginary - summable axis $s$ as to a tangents at at ofz $z= order 1, k =it minus is 1 necessary 1 open ,$ parenthesis\ andquad sufficientit k greater is necessary that 1 closing the Fourier parenthesis and sufficient s and in such that the Fourier sa eries way he offounderies $ the of f relationship $f on $K$K betweenconverges converges the Landauto s at hyphento the $ same Wiener s $ point at case the . open same parenthesis point k = . infinity closing parenthesis and the LittlewoodKaramata hyphen conj Schmidt ectured case open the parenthesis general problem k = 1 closing by replacing parenthesis periodK with The a general parabola case was solved by KaramataKaramata conjhaving quoteright ectured the s studentimaginary the Avakumovi general axis as problem acute-c a tangent to bythe of power replacing order of 2k 1− in1( $k his >K thesis1) $and with period in such a parabola havingAmong thea articles way imaginary he outside found axis the the scop as relationship e a of tangent Tauberian between of theorems order the comma Landau $ k one− should - Wiener1 ( case k (k >= ∞)1and )$ and in such select thosethe which Littlewood are cit ed - and Schmidt which represent case (k new= 1). contributionsThe general in case proofs was solved by 21 \noindentand ideas periodKaramataa way .. he By found’ estimating s student the series Avakumovi relationship with real commac´ betweenin hispositive thesis the and . Landaumonotone− membersWiener comma case $( k = \ infty ) $KaramataAmong arrived t the o the articles general outsidetheorem of the Jensen scop on e convexof Tauberian functions theorems period , one should select andThis the work Littlewoodthose was cit which ed by− Hardy areSchmidt cit comma ed and case Littlewood which $ ( represent and k P acute-o = new 1 lya contributions to ) the power. $ ofThe in 26 proofs periodgeneral and .. Although case ideas was Karamata solved by thought that. these By authors estimating were theseries first with ones real to publish , positive this generalization and monotone members , Karamata \noindentof Jensen quoterightarrivedKaramata t s o theorem ’ the s generalstudent comma theorem Avakumovi they thought of Jensen that $ \ theacute on proof convex{c} andˆ{ functions the2 formulation 1 } .$ inb elong his thesis . 26 t o KaramataThis open work square was bracket cit ed 30 by closing Hardy square , Littlewood bracket period and F-r P subo´lya om Cauchy. Although hyphen Jensen Karamata quoteright s theorem he\noindent derives commaAmongthought as an the that almost articles these authors outside were the the scop first e ones of Tauberian to publish thistheorems generalization , one should of selectevident those consequenceJensen which ’ s comma theorem are several cit , they ed ofMercer and thought which quoteright that represent the s theorems proof new and of elementary contributions the formulation nature open in b elong proofs square t bracket o 25 closing and i d e a s . \quad By estimating series with real , positive and monotone members , square bracketKaramata comma [30]. F − rom Cauchy - Jensen ’ s theorem he derives , as an almost Karamataopen squareevident arrived bracket consequence 26 t closing o the square general , several bracket theorem ofperiod Mercer .. of In ..Jensen’ s open theorems square on convex bracket of elementary 63 functions closing nature square . bracket [ 25 ] comma .. with .. the precision, .. [ and 26 ] .. . the knowledge In [ 63 .. of ] , analytic with t ools the .. so precision and the knowledge of \noindentcharacteristicanalyticThis of him work t comma ools was he cit so derives characteristic ed by several Hardy theorems of , him Littlewood on , singular he derives trigonometric and several P $ \ theoremsacute{o} on singularlya ˆ{ 26 } . $ \quadintegralsAlthough periodtrigonometric .. Karamata This work integrals was cit ed . in the This theory work of was singular cit ed integrals in the of theory Calder of acute-o singular n integrals thoughtand Zygmund thatof Calder comma these Actao´ authorsn andMath Zygmund period were 88 the open , Acta first parenthesis Math ones . 1 88 to 952 ( publish 1closing 952 ) parenthesis 85 this - 139 generalization 85 . hyphen 1 39 period ofBefore Jensen theBefore World ’ s theorem War the IIWorld comma , Warthey .. in II thought .. , 1939 in .. approximately that 1939 the approximately proof comma and .. Karamata the , formulation Karamata felt .. that felt b elong that toKaramatahe was t iredhe ofwas such $[ t ired kind 30 of of worksuch ] and kind . of of such F− workr mathematics{ andom of}$ such period Cauchy mathematics .. He− wasJensen not . ’ He s theorem was not he derives , as an almost only a mathematicianonly a mathematician of value and of of good value reputation and of commagood reputation but he also had, but a he also had a \noindent evident consequence , several of Mercer ’ s theorems of elementary nature [ 25 ] , 2 6 sub G26 periodG. H H . Hardy period , JHardy . E . Littlewood comma J period , G . P Eo´ periodlya , Inequalities Littlewood , Cambridge comma G periodUniversity P acute-o Press , lya 1 952 comma . Inequalities comma[ 26 Cambridge ] . \quad UniversityIn \quad Press[ comma 63 ] , \quad with \quad the precision \quad and \quad the knowledge \quad of analytic t ools \quad so characteristic1 952 period of him , he derives several theorems on singular trigonometric i n t e g r a l s . \quad This work was cit ed in the theory of singular integrals of Calder $ \acute{o} $ n and Zygmund , Acta Math . 88 ( 1 952 ) 85 − 1 39 .

\noindent Before the World War II , \quad in \quad 1939 \quad approximately , \quad Karamata felt \quad that he was t ired of such kind of work and of such mathematics . \quad He was not

\noindent only a mathematician of value and of good reputation , but he also had a

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This i s the way teaching Everythingstarted and has the old vanished fashioned as scientific if it knowledge never existedrefreshedunknown period ....\quad .. ForMathematics the t each hyphen has dis − appeareding of the Theory , as well of Complex as his F-u connections sub nctions comma abroad Karamata , his prepared students a small , comma his t but eaching . \quad In the years 1Everything 94 1 − 1945 has he vanished didn ’ ast even if it neverlook at existed the works . heMathematics had started has , dis\quad - the original commaappeared introduction , as well period as his .. This connections is the book abroad .. quotedblright , his students .. 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Theor routine yan dPrac with t-i c old eo fth habits , \quad and the new order and- modern sity destroyed development , without of lit life erature and , science without . former\quad friendsThis , i new s the ideas way about teaching t Stieltj es Integraleaching quotedblright , about .. science open square ; bracketthe earlier 88 closing routine square with bracket old habits comma , a model and the for new an introduction t o realstarted analysis and period the .. Many old fashioned scientific knowledge refreshed . \quad For the t each − ing of theorder Theory andmodern of Complex development $ F−u of{ lifen c t iand o n s science , }$ . Karamata This i s prepared the way teaching a small , but examples instarted this t extbook and the showed old fashioned his knowledge scientific and the knowledge wealth of all refreshed parts . For the t each - originalof real analysis , introduction period .. It is a . pity\quad that heThis didn is quoteright the book t continue\quad work’’ on\quad such kindCompl of $ e−x $ N \quad umber ’ ’ \quad [ 89 w $ i ˆ{ h−ting}$ of\ thequad Theoryman of Complex F − unctions, Karamata prepared a small , but original examples, and introduction geometric . constructions This is the book avoiding ” Compl classicale − x N notions umber− r ” e a l [ and89 w ih−t imaginaryman part examples . \quad andIn less geometric than constructions weeks he wrote avoiding : \quad classical’’ \quad notionsTheor - real yan and dPrac $ t−i $ c eo f t h imaginary part . In less than weeks he wrote : ” Theor yan dPrac t − i c Stieltj eseo Integral fth Stieltj ’’ es\ Integralquad [ ” 88 ] [ 88 , a ] , model a model for for an an introduction introduction t t o o real real analysis analysis . \quad Many examples. in this Many t examples extbook showed in this t his extbook knowledge showed and his knowledge the wealth and of the all wealth parts of all of real analysisparts of real . \quad analysisIt . is a It pity is a pity that that he he didn didn ’ ’ t t continue continue work work on on such such kind kind of of Jovan Karamata .. 1 9 \noindentt extbooksJovan period Karamata \quad 1 9 Finally comma he returned t o scientific work comma after a quarter of a century comma to the \noindenttheory of regularlyt extbooks varying . functions comma but with new students period .. The preceding p eriod .. of his works was .. ended period .. After .. 1 946 comma .. Karamata .. and his .. students \noindenthave finishedFinally the last ,chapters he returned of this theory t o comma scientific in particular work it , s applications after a quarter period of a century , to the theoryWe give of credit regularly to Karamata varying for creating functions a circle of , mathematicians but with new which students . \quad The preceding pgathered e r i o d \ ..quadJovan around Karamataof his his ideas works period1 9 was .. His\quad merit ..ended i s .. even . \quad bigger becauseAfter \ ..quad of the1 bad 946 , \quad Karamata \quad and h i s \quad students haveconditions finishedt for extbooks scientific the last .work semicolon chapters he of has this also merit theory for the , scientific in particular work which it s applications . Wehas give only startedcreditFinally t o to , develop he Karamata returned and finally t for o scientific merit creating for workthe a appearance ,circle after a ofquarter Yugoslav mathematicians of a century , which to the theory gatheredmathematicsof\quad regularly.. onaround the international varying his ideasfunctions .. stage . ..\ ,quad of but that withHis .. t imenewmerit period students\quad .. Karamata .i s The\quad himself precedingeven bigger p eriod because \quad o f the bad conditionsand his studentsof hisfor marked works scientific was this development ended work ; . periodhe After has ..Alj also 1 an 946 meritcaron-c , ifor Karamata c-acute the and scientific Karamata and his in work open students whichsquare bracket 1 1 2 closinghas only squarehave started bracket finished t o the develop last chapters and finally of this merit theory for , in particularthe appearance it s applications of Yugoslav . 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trix into[ank an] asymptotically \noindentequivalenttunder sequence o transform precise period every conditions slowly varying on the sequence function into $ \ anvarepsilon asymptotically( equivalent t ) ,$ followsfromthis . In .. opensequence square bracket . 1 30 closing square bracket comma .. Baj caron-s anski and Karamata gave a new precise analysis ..The of the firstIn variant [ 1 of30 ] regularly , Baj sˇ varyinganski and sequences Karamata was gave introduced a new precise by Schmidt analysis of (notion 1 925 of ) generalthe\quad notion orderin of theof growth general article in the ordertransfinite mentioned of growth problem b in efore the period transfinite , ..\quad This iand s problem an old in such . a This way i he s an has started Karamataand a wellold ’known s and theory problem a well , which\ knownquad originatesfor problem which from which Hardy the finaloriginates comma form .. but from with , \quad Hardy a newsp , ecially but with for a new the sequences , wasunderstanding givenunderstanding by period Karamata .. In . .. ’ sopen In student square [ 1 28 Mbracket ]. Vuilleumier the 1 28 same closing authors square . \quad bracket studyThere and .. the find, same given the authors principle a sequence study and find the principle$ r of{ nof} equicontinuity= n ˆ{\sigma in the} foundationsL { n } of, regularly n= varying 1functions , 2 , . . . . ,$where $ \{equicontinuityL { n in}\} the foundations$ i s slowlyof regularly varying varying functionssequence period and $ \sigma $ i s any real number , necessary and sufficient conditions are given for the ma − t r i x $ [ a { nk } ] $ t o transform every slowly varying sequence into an asymptotically equivalent sequence .

\noindent In \quad [ 1 30 ] , \quad Baj $ \check{ s } $ anski and Karamata gave a new precise analysis \quad o f the notion of general order of growth in the transfinite problem . \quad This i s an old and a well known problem which originates from Hardy , \quad but with a new understanding . \quad In \quad [ 1 28 ] \quad the same authors study and find the principle of equicontinuity in the foundations of regularly varying functions . 20 .... M period Tomi acute-c \noindentD period Adamovi20 \ h f iacute-c l l M and. Tomi D period $ \ Arandjeloviacute{c} c-acute$ sub comma students of S period Alj an caron-c i c-acute sub comma gave signif hyphen D .icant Adamovi contributions $ \acute t o the{c theory} $ of and regularly D .Arandjelovi varying functions $ comma\acute as{ wellc} as{ , }$ students of S . Alj an $ \otherscheck who{c} completed$ i $ this\acute theory{c in} various{ , } ways$ gaveperiod signif.. Many applications− of icantthese functions contributions are natural t and o the they theory show the of width regularly of this theory varying in differ functions hyphen , as well as othersent fields who period completed this theory in various ways . \quad Many applications of theseKaramata functions20 knew t o arep erfection natural a part and of classical they show analysis the comma width during of thethis time theory in differM . Tomi− c´ ent f i e l d s . of it s rise andD also . Adamovi during thec´ tand ime of D it . s Arandjelovi decline periodc´ .., students He knew not of onlyS . Alj basic an cˇ i c´, gave signif ideas comma-icant .. but .. contributions also the whole t t echnique o the theory of work of comma regularly .. methods varying .. and functions procedures , as as well as Karamatawell as the knewothers lit erature t who o p of completed erfection that t ime period this a part theory .. His of students in classical various partially ways analysis continued . 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Karamata they spread were not left the even range Belgrade in the of University investigations in 1 953period , . .. and At This that most t ime ofwas the his called students then scatteredMathematicalKaramata all Institute over ’ had the s circle no world p ermanent . . In Besides the positions short that comma period , butthey , afterit was have 1 more 929 never like , for a b about een in less the than centers 1 ofclub investigations with meetings0 years , once , and a such week shortly periodas the after .. MostDepartment 1946 of Karamata , this of quoterightcircleMathematics was s made students of , the were enlarged University looking and then offor Belgrade positionsslowly abroad , \quad lost period powerand .. They they . 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On CeREFERENCES r-t ain Limits Similar to Definite Integrals period Doctoral dissertation \ hspace ∗{\ f i l l } [ 1 ] \quad0 Sur l $ ’ {\acute{e}}$ valuation des limites se rattachant aux suites period Belgrade comma[ 1 ] Sur l e´ valuation des limites se rattachant aux suites a` doubles entr e´ es . Glas Srpske $ \1grave 926 period{a} $ p period doublesKralj 65 period evske entr Akademije open $ parenthesis\acute , CXX{ ,e SerbianPrvi} $ razred es closing , . 55 Glas , parenthesis 1 926 Srpske , pp . 75 - 1 0 1 . ( Serbian ; French open square bracket 3 closing square bracket .. Sur ce r-t aines limites rattach e-acute es aux int acute-e grales de Stieltjes period\ hspace .. Comptes∗{\ f i l l rendus} Kralj des evske Akademije , CXXsummary) , Prvi razred , 55 , 1 926 , pp . 75 − 1 0 1 . ( Serbian ; French s acute-e ances de l quoteright Acad e-acute mie des Sciences comma Paris comma 1 82 comma 1 9 26 comma pp period \ [ summary[ ) 2 ]\ ] On Ce r − t ain Limits Similar to Definite Integrals . Doctoral dissertation . Belgrade , 833 hyphen 8351 926 period . p . 65 . ( Serbian ) [ 3 ] Sur ce r − t aines limites rattach e´ es aux int e´ grales de Stieltjes . Comptes rendus [ 2 ] \quaddesOn s e´ Ceances $ de r− lt ’ Acad $ aine´ mie Limits des Sciences Similar , Paris , to 1 82 Definite , 1 9 26 , pp Integrals . 833 - 835 . . Doctoral dissertation . Belgrade , 1 926 . p . 65 . ( Serbian )

[ 3 ] \quad Sur ce $ r−t $ aines limites rattach $ \acute{e} $ es aux i n t $ \acute{e} $ grales de Stieltjes . \quad Comptes rendus des s $ \acute{e} $ ances de l ’ Acad $ \acute{e} $ mie des Sciences , Paris , 1 82 , 1 9 26 , pp . 833 − 835 . 22 .... M period Tomi acute-c \noindentopen square22 bracket\ h f i l 4 l closingM . Tomi square $ bracket\acute ..{ Relationc} $ entre les fonctions de r acute-e pa to the power of t-r ition de deux suites d acute-e pendant l quoteright une de l quoteright autre period [ 4Comptes ] \quad rendusRelation des s acute-e entre ances les de fonctions l quoteright Acad de r e-acute $ \acute mie des{e Sciences} pa comma ˆ{ t−r Paris}$ comma ition 1 83 de comma deux 1 suites 9 d 26$ \ commaacute pp{e} period$ pendant 726 hyphen l ’ une de l ’ autre . Comptes728 period rendus des s $ \acute{e} $ ances de l ’ Acad $ \acute{e} $ mie des Sciences , Paris , 1 83 , 1 9 26 , pp . 726 − 728open . square bracket 5 closing square bracket .. Sur une limite inf acute-e rieure du module des z e-acute ros des fonctions analytiques period22 Glas Srpske M . Tomi c´ t−r \ hspaceKraljevske∗{\ f Akademije i l[ l 4} ][ 5Relation comma ] \quad entre CXXVIISur les fonctions une comma limite dePrvi r razredepa´ inf comma $ition\acute de 5 deux 8{ commae suites} $ 1 d rieure 927e´ pendant comma du ppl module ’ une period de l 1des ’ 1 hyphen z $ 1\acute 20 {e} $ periodros des open fonctions parenthesisautre . Comptes Serbian analytiques rendus semicolon des . s French Glase´ ances Srpske de l ’ Acad e´ mie des Sciences , Paris , 1 83 , 1 9 26 , pp . 726 summary closing- 728 . parenthesis \ hspaceopen square∗{\ f i bracket l l } Kraljevske[ 56 ]closingSur square une Akademije limite bracket inf ..e´ , Surrieure CXXVII une du suite module , de Prvi fonctions des z razrede´ ros rationnelles des , fonctions 5 8 et , les1 analytiques d 927 acute-e , pp . veloppementsGlas . 1 1 − suivant1 20 . ( Serbian ; French ces fonctions periodSrpske \ [Glas summary Srpske Kralj ) Kraljevske\ ] evske Akademije Akademije comma , CXXVII CXXVIII , Prvi razred comma , 5 Prvi8 , 1 927 razred , pp comma . 1 1 - 1 59 20 comma . ( Serbian 1 927 ; French comma p period 47 hyphen 79 period open parenthesis Ser hyphen bian semicolon French summary closing parenthesis summary) [ 6 ] \quad Sur une suite de fonctions rationnelles et les d $ \acute{e} $ veloppements suivant ces fonctions . open square bracket[ 6 ] Sur 7 closing une suite square de fonctions bracket rationnelles .. Sur les et suites les d acute-ee´ veloppements quivalentes suivant period ces fonctions Rad Jugoslavenske . Glas akademije Glas Srpske Kralj evske Akademije , CXXVIII , Prvi razred , 59 , 1 927 , p . 47 − 79 . ( Ser − znanosti i umjetnostiSrpske Kralj comma evske 234 Akademije comma , CXXVIII , Prvi razred , 59 , 1 927 , p . 47 - 79 . ( Ser - Razred matemati caron-c ko hyphen prirodoslovnibian ;comma French 71summary comma ) .. 1 928 comma pp period 223 hyphen 245 period .. \ centerline { bian ; French summary ) } open parenthesis[ Serbian 7 ] Sur closing les suites parenthesise´ quivalentes period Bulletin . Rad Jugoslavenske in hyphen akademije znanosti i umjetnosti , 234 , ternationalRazred de l quoteright matemati Acadcˇ ko - acute-e prirodoslovni mie yougoslave , 71 , 1 des 928 sciences , pp . 223 et - des 245 beaux . ( hyphen Serbian arts ) . Bulletin comma in 2 2- comma 1 9 28 [ 7 ] \quad Sur les suites $ \acute{e} $ quivalentes . Rad Jugoslavenske akademije znanosti i umjetnosti , 234 , comma pp periodternational de l ’ Acad e´ mie yougoslave des sciences et des beaux - arts , 2 2 , 1 9 28 , pp . 27 - 34 . ( Razred matemati $ \check{c} $ ko − prirodoslovni , 71 , \quad 1 928 , pp . 223 − 245 . \quad ( Serbian ) . Bulletin in − 27 hyphenFrench 34 period summary open ) parenthesis French summary closing parenthesis ternational de l ’ Acad $ \acute{e} $ mie yougoslave des sciences et des beaux − arts ,22 ,1928 ,pp . open square bracket[ 8 ] 8 closingUne question square de bracket minimum .. Une relative question aux deensembles minimum et sonrelativerappo auxr ensembles−t avec l et’ analyse son rappo . to the power 27 − 34 . ( French summary ) of r-t avec l quoterightComptes rendus analyse du period52econgre` s de l ’ A . F . A . S . , La Rochelle , 1 928 . 2 pages . Comptes rendus[ 9 ] du 5Sur 2 lato moyenne the power arithm of e congre´ tique to des the co powere´ fficients of grave-e d ’ une s s dee´ lrie quoteright de Taylor A . periodMathematica F period A period S [ 8 ] \quad Une question de minimum relative aux ensembles et son $ rappo ˆ{ r−t }$ avec l ’ analyse . period comma( La Cluj Rochelle ) , 1 , 1 comma9 29 , pp 1 . 928 9 9 period- 1 6 . 2 pages period Comptes rendus du $ 5 2 ˆ{ e } congr ˆ{\grave{e}}$ s de l ’A. F .A. S . , La Rochelle , 1928 . 2 pages . open square bracket[ 1 0 9 ] closingTh squaree´ or brackete` mes inverses .. Sur de la sommabilitmoyenne arithme´ I. acute-eGlas Srpske tique Kralj des evskeco e-acute Akademije fficients , d quoteright une s acute-eCXLIII rie de Taylor , period Mathematica [ 9 ] \quad Sur la moyenne arithm $ \acute{e} $ tique des co $ \acute{e} $ fficients d ’ une s open parenthesis Cluj closingPrvi parenthesis razred , 70 , comma 1 931 , pp1 comma . 1 - 24 1 . 9 ( Serbian29 comma ; French pp period summary 9 9 )hyphen 1 6 period $ \acute{e} $ rie de Taylor . Mathematica open square bracket[ 1 1 ] 1 0Th closinge´ or squaree` mes bracket inverses .. deTh sommabilit acute-e or e-gravee´ II . Glas mes inverses Srpske Kralj de sommabilit evske Akademije acute-e , I period Glas ( Cluj ) ,1 ,1929 ,pp.99 − 1 6 . Srpske Kralj evskeCXLIII Akademije , Prvi razred comma , 70 CXLIII , 1 931 , comma pp . 1 2 1 - 146 . ( Serbian ; French summary ) Prvi razred comma[ 1 70 2 ] commaSur ce1 931r − commat ains ” pp period Tauberia 1 hyphen theorem 24 period s ” open d parenthesis M . M Serbian Hard semicolon e French \ hspace ∗{\ f i l l } [ 1 0 ] \quad Th $ \acute{e} $ or $ \grave{e} $ mes inverses de sommabilit summary closingLittlewood parenthesisComptes rendus $ \acute{e} $ I . Glas Srpske Kralj evske Akademije , CXLIII , open square bracketdu 1econgr 1 1 closinge` s dessquare math e´ bracketmaticiens .. Th roumains acute-e , or 1 e-grave 9 29 , mes Cluj inverses . Mathematica de sommabilit ( Cluj acute-e) 3 , 1 II period Glas Srpske Kralj evske930 , pp Akademije . 33 - 48 comma. CXLIII comma \ centerline { Prvi razred , 70 , 1 931 , pp . 1 − 24 . ( Serbian ; French summary ) } Prvi razred comma[ 1 3 ] 70 commaAsymptotes 1 931 des comma courbes pp planes period d 1e´ 2finies 1 hyphen comme 146 enveloppes period open d ’ un parenthesis syst e` me Serbian de droites semicolon French summary closing. Comptes parenthesis rendus du 53econgre` s de l ’ A . F . A . S . , Le Havre 1 929 . 3 pages . [ 1 1 ] \quad Th $ \acute{e} $ or $ \grave{e} $ mes inverses de sommabilit $ \acute{e} $ open square bracket[ 14 ] 1Sommabilit 2 closing squaree´ et bracketfonctionnelles .. Sur lin ce r-te´ aires ains .quotedblrightComptes rendus .. Tauberia du 1econgr .. theoreme` s des math s quotedblright .. d II . Glas Srpske Kralj evske Akademije , CXLIII , .. M period Me´− ..maticiens Hard .. e des .. Littlewood pays slaves Comptes, 1 929 , Varsovie rendus , 1 930 , pp . 2 2 1 - 228 . Prvi razred , 70 , 1 931 , pp . 1 2 1 − 146 . ( Serbian ; French summary ) du 1 to the power[ 1 5 ] of eSur congr une to mode the de power croissance of grave-e r e´ sguli dese math` re des e-acute fonctions maticiens . Mathematica roumains ( Cluj comma ) vol .. . 1 4 9 29 comma .. Cluj period Mathematica, 1 930 , pp . open 38 - 53parenthesis . Cluj closing parenthesis 3 comma \ hspace ∗{\ f i l l } [ 1 2 ] \quad Sur ce $ r−t $ a i n s ’ ’ \quad Tauberia \quad theorem s ’ ’ \quad d \quad M.M \quad Hard \quad e \quad Littlewood Comptes rendus 1 930 comma[16] pp periodU¨ ber 33 hyphen die 48 Hardy period - Littlewoodschen Umkehrungen des Abelschen St a¨ open squaretigkeitssatzes bracket 1 . 3 closingMathematische square bracket Zeitschrift .. , Asymptotes B . 32 , H . 2 des , 1 9courbes 30 , pp planes . 31 9 - d 320 acute-e . finies comme enveloppes d du $ 1 ˆ{ e } congr ˆ{\grave{e}}$ s des math $ \acute{e} $ maticiens roumains , \quad 1 9 29 , \quad Cluj . Mathematica ( Cluj ) 3 , quoteright un syst[ 1 e-grave 7 ] Sur me dele principe droites periodde maximum des fonctions analytiques et son application au th e´ or e` 1 930 , pp . 33 − 48 . e e` Comptes rendusme de du d ’ 5 Alembe 3 to ther power− t. Comptes of e congr rendus to the du power54 congr of grave-es de l ’ sA de . F l . quoteright A . S . , Alger A period , 1 930 F , pp period . A period S period comma29 Le - 31 Havre . 1 929 period 3 pages period [ 1 3 ] \quad Asymptotes des courbes planes d $ \acute{e} $ finies comme enveloppes d ’ un syst open square bracket[ 1 8 ] 14Quelques closing square th e´ bracketor e` mes .. Sommabilit d ’ inversion acute-e relatifs et aux fonctionnelles int e´ grales lin et e-acute aux s airese´ ries period Comptes $ \grave{e} $ me de droites . rendus du 1 to(1 theere` pa powert−r ie of ) e . congrBulletin to the de power mathematiques of grave-e et s de des physique math e-acute de l ’ Ecole hyphen Polytechnique de Bucarest , Comptes rendus du $ 5 3 ˆ{ e } congr ˆ{\grave{e}}$ s de l ’A.F.A. S . , LeHavre1929 . 3 pages . maticiens des3 , 1pays 932 slaves , no . 1 comma , fasc . 1 7 929 , pp comma . 1 - 1 0 Varsovie . comma 1 930 comma pp period 2 2 1 hyphen 228 period open square bracket[ 1 9 ] 1 5Neuer closing square Beweis bracket und .. Sur Verallgemeinerung une mode de croissance einiger r acute-e Tauberian guli e-grave - S rea¨ destze fonctions . period [ 14 ] \quad Sommabilit $ \acute{e} $ et fonctionnelles lin $ \acute{e} $ aires . Comptes rendus du MathematicaMathematische open parenthesis Zeitschrift Cluj closing B . 33 parenthesis , H . 2 , 1 931 vol , period pp . 294 4 comma - 2 99 . 1 930 comma $ 1 ˆ{ e } congr ˆ{\grave{e}}$ s des math $ \acute{e} − $ pp period 38 hyphen[ 20 ] 53Neuer period Beweis und Verallgemeinerung der Tauberschen S a¨ tze , welche die Laplacesche maticiens des pays slaves , 1 929 , Varsovie , 1 930 , pp . 2 2 1 − 228 . open squareund bracket Stieltjessche 1 6 closing Transformation square bracket betreffen U-dieresis . Journal ber f u¨ ..r die die .. reine Hardy und hyphenangewandte Littlewoodschen Mathematik , .. B Umkehrungen . .. des .. Abelschen1 64 .. , StH . a-dieresis 1 , 1 931 , tigkeitssatzes pp . 27 - 39 . period [ 1Mathematische 5 ] \quad ZeitschriftSur une commamode de B period croissance 32 comma r H period$ \acute 2 comma{e} $ 1 9 30 g u comma l i $ pp\grave period{e 31} 9$ hyphen re des 320 period fonctions . Mathematica ( Cluj ) vol . 4 , 1 930 , ppopen . 38 square− 53 bracket . 1 7 closing square bracket .. Sur le principe de maximum des fonctions analytiques et son application au th acute-e or e-grave me $de [ d quoteright 1 6 Alembe ] \ddot r-t sub{U} period$ ber Comptes\quad rendusd i e du\quad 54 toHardy the power− Littlewoodschen of e congr to the power\quad of e-graveUmkehrungen s de l \quad des \quad Abelschen \quad St quoteright$ \ddot{ Aa} period$ tigkeitssatzes F period A period S . period comma Alger comma 1 930 comma pp period 29 hyphen 31 period Mathematischeopen square bracket Zeitschrift 1 8 closing square , B bracket . 32 , .. H Quelques . 2 , th 1 acute-e 9 30 or, ppe-grave . 31 mes 9 d− quoteright320 . inversion relatifs aux int acute-e grales et aux s e-acute ries open parenthesis 1 to the power of e-grave re pa to the power of t-r ie closing parenthesis period[ 1 7 ] \quad Sur le principe de maximum des fonctions analytiques et son application au th $ \Bulletinacute{ dee} mathematiques$ or $ \grave et de physique{e} $ deme l quoteright Ecole Polytechnique de Bucarest comma 3 comma de1 932d ’ comma Alembe no period $ r−t 1 comma{ . } fasc$ period Comptes 7 comma rendus pp period du $ 1 hyphen 54 ˆ{ 1e 0} periodcongr ˆ{\grave{e}}$ sde l ’A.F.A.S. ,Alger ,1930 ,pp. 29 − 31 . open square bracket 1 9 closing square bracket .. Neuer .. Beweis .. und .. Verallgemeinerung .. einiger .. Tauberian hyphen[ 1 8 S ] a-dieresis\quad Quelques tze period .. th Mathematische $ \acute{e} $ or $ \grave{e} $ mes d ’ inversion relatifs aux int $ \Zeitschriftacute{e} B$ period grales 33 comma et aux H period s $ 2 comma\acute 1{ 931e} comma$ r i e pp s period $ ( 294 1 hyphen ˆ{\grave 2 99 period{e} re } pa ˆ{ t−r }$ i eopen ) . square bracket 20 closing square bracket .. Neuer Beweis und Verallgemeinerung der Tauberschen S dieresis-a tze commaBulletin welche de die mathematiques Laplacesche et de physique de l ’ Ecole Polytechnique de Bucarest , 3 , 1932und Stieltjessche ,no . 1 Transformation , fasc . 7 betreffen ,pp . period1 − 1 Journal 0 . f u-dieresis r die reine und angewandte Mathematik comma B period 1 64 comma H period 1 comma 1 931 comma pp period 27 hyphen 39 period [ 1 9 ] \quad Neuer \quad Beweis \quad und \quad Verallgemeinerung \quad e i n i g e r \quad Tauberian − S $ \ddot{a} $ t z e . \quad Mathematische Zeitschrift B . 33 , H . 2 , 1 931 , pp . 294 − 2 99 .

[ 20 ] \quad Neuer Beweis und Verallgemeinerung der Tauberschen S $ \ddot{a} $ tze , welche die Laplacesche und Stieltjessche Transformation betreffen . Journal f $ \ddot{u} $ r die reine und angewandte Mathematik , B . 1 64 ,H . 1 , 1 931 , pp . 27 − 39 . Jovan Karamata .... 23 \noindentopen squareJovan bracket Karamata 2 1 closing square\ h f i l bracket l 23 .. Sommablit acute-e et convergence period Glas Srpske Kraljevske Akademije comma CXLVI comma Prvi razred comma \ hspace72 comma∗{\ f 1 i l 931 l } [ comma 2 1 ] pp\quad periodSommablit 59 hyphen 67 $ period\acute open{e} parenthesis$ et convergence Serbian semicolon . Glas French Srpske summary Kraljevske closing Akademije , CXLVI , Prvi razred , parenthesis \ centerlineopen square{ bracket72 , 1 22 931 closing , pp square . 59 bracket− 67 .. . Sur ( le Serbian rappo to the; French power of summary r-t entre les ) convergences} d quoteright une suite de fonctions et de leurs moments [ 22avec ] application\quadJovanSur Karamata grave-a le l$ quoteright rappo ˆ{ inversionr−t }$ des entre proc to les the power convergences of e-acute d d acute-e ’ une s suite de sommabilit de23 fonctions e-acute sub et de leurs moments periodavec Studia application Mathematica[ 2 1 ] Sommablit $ comma\grave T{e´a periodet} convergence$ l ’ inversion . Glas Srpske des Kraljevske $ proc Akademije ˆ{\acute , CXLVI{e ,}} Prvi$ razred d , $ \acute{e} $ s de3 comma sommabilit 1 931 comma $ \ ppacute period{72e} 68 , 1 hyphen 931{ . , pp}$ 76 . 59 period Studia - 67 . ( Serbian Mathematica ; French summary , T . ) r−t 3open , 1931 square ,pp bracket[ 22 ] . 68 23Sur− closing le76rappo . squareentre bracket les convergencesU-dieresis ber d ’ einen une suite Satz de von fonctions Vijayaraghavan et de leurs moments period Mathematische e´ Zeitschrift commaavec application B period 34a` commal ’ inversion H period des 5proc commad 1e´ 932s de comma sommabilit e´. Studia Mathematica , T . 3 , 1 $pp [ period 23931 737 ] , hyphen pp .\ddot 68 740 - 76{U period . } $ ber einen Satz von Vijayaraghavan . Mathematische Zeitschrift , B . 34 , H . 5 , 1 932 , ¨ ppopen . 737 square− bracket740[23] .U 24ber closing einen Satz square von bracket Vijayaraghavan .. Directions . Mathematische des tangentes Zeitschrift en relation , B . avec34 , H l .quoteright 5 , 1 932 , aire de surface period Comptespp rendus. 737 - 740 du 5. 5 to the power of e e \ hspacecongr to∗{\ thef i power l l }[[ 24 of 24 ] grave-e ] Directions\quad s de lDirections des quoteright tangentes A en period des relation tangentes F period avec l ’ A aire period en de relationsurface S period . Comptes comma avec rendusNancy l ’ aire du comma55 de 1 surface9 31 comma . Comptes rendus du e` pp$ 5 period 5 44 ˆ{ hyphene }$ 46 period congr s de l ’ A . F . A . S . , Nancy , 1 9 31 , pp . 44 - 46 . open square bracket[ 25 ] 25Sur closing quelques square bracket inversions .. Sur .. quelques d ’ une .. inversions proposition .. d quoteright de Cauchy une .. proposition et .. de .. e´ Cauchy\ centerline .. et ..leurs{ leurs$ congr..g g ton the ˆe´{\ralisations powergrave ofacute-e{ . e}}T oˆ$ nhoku e-acute sde Mathematical ralisations l ’A.F.A.S. Journal period , Vol . 36 , 1,Nancy 932 , pp . ,1931 22 - 28 . ,pp. 44 − 46 . } e´ T o-circumflex[ hoku 26 ] MathematicalSur un th e´ Journalor e` me comma de Mercer Vol periodet ses 36g comman e´ ralisations 1 932 comma . Glas pp Srpske period Kralj 22 hyphen evske 28 period [ 25open ] square\quadAkademije bracketSur \ 26,quad closingquelques square bracket\quad .. Suri n v un e rth s i o acute-e n s \quad or e-graved ’ me une de\ Mercerquad etproposition ses g to the power\quad of acute-ede \quad Cauchy \quad et \quad l e u r s \quad n$ e-acute g ˆ{\ ralisationsacute{e}} periodCXLVI$ Glas n , Prvi $Srpske razred\acute Kralj , 72{e , evske} 1 9$ 31 Akademije ,ralisations pp . 1 1 3 comma - 1 20 . . ( Serbian ; French summary ) TCXLVI $ \hat comma{o}[ 27 Prvi$ ] hoku razredApplication Mathematical comma 72 comma de Journal 1 quelques 9 31 comma , Vol th pp period.e´ or 36 ,1e` 1 1mes 3 932 hyphen , d pp 1’ inversion20 . period 22 − open28a` parenthesis . la Serbian semicolon Frenchsommabilit summarye´ closingexponentielle parenthesis . Comptes rendus des s e´ ances de l ’ Acad e´ mie des Sciences , Paris \ hspaceopen square∗{\,f 1 i bracket 93l l } , 1[ 931 26 27 , closing ]pp\ .quad 1 1 square 56Sur - 1 bracket 1 un 5 8 . th .. Application $ \acute{ ..e de} $ .. quelques or $ ..\grave th acute-e{e} or$ e-grave me de mes Mercer .. d quoteright et ses inversion$ g ˆ{\ ..acute grave-a[{ 28e ..}} ] la ..$Remarks sommabilit n $ on\acute a e-acute theorem{e ..} of exponentielle$ D . ralisations V . Widder period . .Proceedings Glas Srpske of the Kralj Natural evske Academy Akademije of , Comptes rendusSciences des USA s acute-e , 1 8 , 1 ances 932 , de pp l . quoteright 406 - 408 . Acad e-acute mie des Sciences comma Paris comma 1 93 comma 1 931 r−t e´ comma\ centerline pp period{CXLVI[29] 1 1 56Rapo hyphen, Prvi razredentre ,les 72 , limites 1 9 31 ,d ’pp oscillations . 1 1 3 − des1 20proc . (d Serbiane´ s ; de French summary ) } 1 1 5 8 periodsommation d ’ Abel et de Ces a` ro . Publications math e´ matiques de l ’ Universit e´ de [ 27open ] square\quadBelgrade bracketApplication , t28 . closing1 , 1 932 square\ ,quad pp . bracket 1de 1 9\ -quad 1.. 24 Remarks . quelques on a theorem\quad ofth D period $ \acute V period{e} ..$ Widder or period $ \grave Proceedings{e} $ ofmes the\ Naturalquad d Academy ’[ 30 inversion ] ofSur une\ inquade´ galit$ \e´graverelative{ auxa} fonctions$ \quad convexesl a \ .quadPublicationssommabilit math e´ matiques $ \acute de {e} $ \quad exponentielle . ComptesSciences USA rendusl ’ Universit comma des 1e´ 8sde comma Belgrade $ \acute 1 932 , t .{ comma 1e} , 1$ 932 pp ances , pp period . 145 de 406 - 148 l hyphen .’ Acad 408 period$ \acute{e} $ mie des Sciences , Paris , 1 93 , 1 931 , pp . 1 1 56 − 1open 1 5 square 8 . bracket[ 31 ] 29 closingQuelques square th e´ bracketor e` mes Rapo de nature to the tauberienne power of r-t . ..Studia entre Mathematica .. les .. limites , T .. . 4 d , quoteright 1 933 , oscillations .. des proc topp the . power4 - 7 . of e-acute d acute-e s .. de sommation .. d quoteright Abel et .. de [ 28 ] \quad Remarks on a theorem ofe´ D . V . \quad Widder . Proceedingse´ of the Natural Academy of Ces a-grave ro[ 32period ] PublicationsUn th e´ or mathe` me e-acuteg n matiquese´ ral d ’ de inversion l quoteright des Universitproc d acute-ee´ s de sommabilit de Belgradee´ comma. t period 1Sciences comma 1 932 USAVerhandlungen comma , 1 pp 8 period , des 1 Internationaler 932 , pp . mathematiker 406 − 408 Kongress . , Z u¨ rich , 1 932 , II , pp . 147 - 149 . 1 1 9 hyphen 1[ 33 24 ] periodUn th e´ or e` me sur les coefficients des s e´ ries de Taylor $open [ square 29. bracket ]Glas Rapo 30 Srpske closing ˆ{ r− Kralj squaret } evske$ bracket\quad ..entr Sur e une\quad in acute-el e s galit\quad e-acutel i m relative i t e s \quad aux fonctionsd ’ oscillations convexes period\quad des Publications$ proc ˆ{\ mathacuteAkademije acute-e{e}} matiques , CLIV$ d , Prvi de $ \ razredacute ,{ 77e} , 1$ 933 ,s pp\quad . 79 - 9de 1 . (sommation Serbian ) . Bulletin\quad ded l ’ ’Acad Abele´ mie et \quad de Cesl quoteright $ \grave Universitroyale{a} $ serbe acute-e ro , S . dee´ Publicationsrie Belgrade A : Sciences comma math math t periode´ matiques $ 1\ commaacute , t . 1{ ,e 932 1} 933$ comma , ppmatiques . pp 85 -period 89 . de ( 145French l hyphen ’ sum Universit - 148 period $ \acute{e} $ deopen Belgrade square bracket , t . 31 1 closing , 1 932 square , pp bracket . .. Quelques th acute-e or e-grave mes de nature tauberienne period Studia Mathematica1 1 9 − 1 comma 24 . T period 4 comma 1 933 comma pp periodmary) 4 hyphen 7 period open square bracket¨ 32 closing square bracket .. Un th acute-e or e-grave me g to the power of acute-e n e-acute ral d [ 30 ] \quad [34]SurU uneber diein O $ - Inversionss\acute{e}a¨ tze$ der g a Limitierungsverfahren l i t $ \acute{e} . $Mathematische relative aux Zeitschrift fonctions , convexes . Publications math quoteright inversionB . 37 , des H . proc 4 , 1 933to the , pp power . 582 of- 588 acute-e . d e-acute s de sommabilit acute-e sub period Verhandlungen des $ \Internationaleracute{e} $ mathematiker matiques de Kongress comma Z u-dieresis rich comma 1 932 comma II comma pp period 147 hyphen 149 l ’ Universit[ 35 ] $ \Thacutee´ or{e} e`$mes de sur Belgrade la sommabilit , te´ exponentielle . 1 , 1 932 et d, ’ autres pp . sommabilit 145 − 148e´ s . s ’ y period rattachant . Comptes rendus du 2econgre` s des math e´ maticiens roumains , 1 932 , Turnu Severin open square bracket 33 closing square bracket .. Un .. th acute-e or e-grave me .. sur les .. coefficients .. des .. s acute-e \ hspace ∗{\.f i Mathematica l l } [ 31 ] (\ Clujquad ) , 9Quelques , 1 935 , ppth . 1 64 $ - 1\ 78acute . {e} $ or $ \grave{e} $ mes de nature tauberienne . Studia Mathematica , T . 4 , 1 933 , pp . 4 − 7 . ries .. de .. Taylor[ 36 period ] Compl .. Glas ..e´ Srpskement Kralj au evske premier th e´ or e` me de la moyenne . AkademijeGlas comma Srpske CLIV comma Kralj evske Prvi razred Akademije comma , CLIV77 comma , Prvi 1 933 razredcomma , pp 77 period , 1 79 933 hyphen , pp 9 . 1 period open parenthesis[ 32 ] \quad SerbianUn closing th $parenthesis\acute{ periode} $ Bulletin or $ de\grave l quoteright{e} $ Acad me acute-e $ g mie ˆ{\acute{e}}$ n $ \acute{e} $ ral d ’ inversion1 1 9 - 144 . des ( Serbian $ proc ) . ˆ{\ Bulletinacute{e de}}$ d $ \acute{e} $ s de sommabilit $ \acute{e} { . }$ royale serbe commal ’ Acad Se´ acute-emie royale rie A serbe : Sciences , S e´ rie math A : Sciencese-acute matiquesmath e´ matiques comma , t t period . 1 , 1 1 933 comma , pp . 1 97 933 - 1 comma6 . pp period 85Verhandlungen hyphen 89 period des open parenthesis French sum hyphen Internationaler( French summary mathematiker ) Kongress , Z $ \ddot{u} $ rich , 1932 , II , pp . 147 − 149 . mary closing parenthesis[ 37 ] Sur une mode de croissance r e´ guli e` re . Th e´ or e` mes fondamentaux . open squareBulletin bracket de 34 la closing Soci e´ t squaree´ math brackete´ matique U-dieresis de France ber ,die t . O 6 1hyphen , 1 9 33 Inversionss , pp . 55 - 62 a-dieresis . tze der Limitierungsver- fahren\ hspace period∗{\ ..f i Mathematische l l } [ 33 ] \quad ZeitschriftUn \ commaquad th $ \acute{e} $ or $ \grave{e} $ me \quad sur l e s \quad coefficients \quad des \quad s $ \Bacute period{e 37} comma$ r i e H s period\quad 4 commade \quad 1 933Taylor comma pp . \ periodquad 582Glas hyphen\quad 588Srpske period Kralj evske open square bracket 35 closing square bracket .. Th acute-e or e-grave mes sur la sommabilit acute-e exponentielle et d quoteright\ hspace ∗{\ autresf i l sommabilitl }Akademije e-acute , CLIV s s quoteright , Prvi y razredrattachant , period 77 , 1 933 , pp . 79 − 9 1 . ( Serbian ) . Bulletin de l ’ Acad $ \Comptesacute{e rendus} $ du mie 2 to the power of e congr to the power of grave-e s des math e-acute maticiens roumains comma 1 932 comma Turnu Severin period \ hspaceMathematica∗{\ f i l open l } royale parenthesis serbe Cluj , closing S $ parenthesis\acute{e comma} $ rie 9 comma A : 1 Sciences 935 comma math pp period $ \ 1acute 64 hyphen{e} 1$ 78 matiques period , t . 1 , 1 933 , pp . 85 − 89 . ( French sum − open square bracket 36 closing square bracket .. Compl acute-e ment .. au .. premier .. th e-acute or grave-e me .. de .. la\ [ .. mary moyenne ) period\ ] .. Glas .. Srpske .. Kralj evske Akademije comma .. CLIV comma .. Prvi .. razred comma .. 77 comma .. 1 933 comma .. pp period .. 1 1 9 hyphen 144 period .. open parenthesis Serbian closing parenthesis period .. Bulletin .. de $l [ quoteright 34 Acad ] \ acute-eddot{U mie} $ royale ber serbe d i e comma O − Inversionss S e-acute rie A : $ Sciences\ddot{ matha} $ acute-e tze matiques der Limitierungsverfahren comma t period 1 . \quad Mathematische Zeitschrift , commaB. 37 1 933 ,H. comma 4 pp ,1933 period 97 ,pp hyphen . 582 1 6 period− 588 . open parenthesis French summary closing parenthesis [ 35open ] square\quad bracketTh $ 37\ closingacute{ squaree} $ bracket or $ .. Sur\grave une{ modee} $ de croissancemes sur r la acute-e sommabilit guli e-grave $ re\ periodacute ..{e Th} acute-e$ exponentielle et d ’ autres sommabilit or$ e-grave\acute mes{e} fondamentaux$ s s ’ y period rattachant .. Bulletin . de la ComptesSoci acute-e rendus t e-acute du math $ 2 acute-e ˆ{ e matique} congr de France ˆ{\grave comma{e t}} period$ 6 s 1 des comma math 1 9 33 $ comma\acute pp{e period} $ 55 maticiens hyphen 62 roumains , 1 932 , Turnu Severin . periodMathematica ( Cluj ) , 9 , 1 935 , pp . 1 64 − 1 78 . open square bracket 38 closing square bracket .. Sur les th acute-e or e-grave mes de nature tauberienne period Comptes rendus[ 36 ]des\quad s acute-eCompl ances de $ l\ quoterightacute{e} Acad$ ment e-acute\quad mie au \quad premier \quad th $ \acute{e} $ or $ \desgrave Sciences{e} comma$ me Paris\quad commade \ 1quad 97 commal a \ 1quad 933 commamoyenne pp period . \quad 888 hyphenGlas \ 890quad periodSrpske \quad Kralj evske Akademije , \quad CLIV , \quad Prvi \quad razred , \quad 77 , \quad 1 933 , \quad pp . \quad 1 1 9 − 144 . \quad ( Serbian ) . \quad B u l l e t i n \quad de

l ’ Acad $ \acute{e} $ mie royale serbe , S $ \acute{e} $ rie A : Sciences math $ \acute{e} $ matiques , t . 1 , 1 933 , pp . 97 − 1 6 . ( French summary )

[ 37 ] \quad Sur une mode de croissance r $ \acute{e} $ g u l i $ \grave{e} $ re . \quad Th $ \acute{e} $ or $ \grave{e} $ mes fondamentaux . \quad Bulletin de la Soci $ \acute{e} $ t $ \acute{e} $ math $ \acute{e} $ matique de France , t . 6 1 , 1 9 33 , pp . 55 − 62 .

[ 38 ] \quad Sur l e s th $ \acute{e} $ or $ \grave{e} $ mes de nature tauberienne . Comptes rendus des s $ \acute{e} $ ances de l ’ Acad $ \acute{e} $ mie des Sciences , Paris , 1 97 , 1 933 , pp . 888 − 890 . [ 38 ] Sur les th e´ or e` mes de nature tauberienne . Comptes rendus des s e´ ances de l ’ Acad e´ mie des Sciences , Paris , 1 97 , 1 933 , pp . 888 - 890 . 24 .... M period Tomi acute-c \noindentopen square24 bracket\ h f i l 39 l M closing . Tomi square $ bracket\acute ..{ Weiterfc} $ u-dieresis hrung der N period .. Wienerschen Methode period .. Mathematische Zeitschrift comma B period 38 comma H period [ 395 comma ] \quad 1 934Weiterf comma pp $period\ddot 701{u hyphen} $ hrung 708 period der N . \quad Wienerschen Methode . \quad Mathematische Zeitschrift , B . 38 , H . 5open , 1 square 934 , bracket pp . 40 701 closing− 708 square . bracket .. Einige weitere Konvergenzbedingungen der Inversionss dieresis-a tze der Limitierungsverfah hyphen [ 40ren ]period\quad PublicationsEinige mathweitere acute-e Konvergenzbedingungen matiques de l quoteright Universit der Inversionss e-acute de Belgrade $ \ddot comma{a} t period$ tze 2 comma der Limitierungsverfah 1 − 933ren comma . Publications pp24 period 1 hyphen math 1 6 $ period\acute{e} $ matiques de l ’ Universit $ \acuteM . Tomi{e} c´$ de Belgrade , t . 2 , 1 933 , pp . 1 − 1 6 . open square bracket[ 39 ] 41Weiterf closing squareu¨ hrung bracket der N . .. L Wienerschendieresis-o sung Methode der Aufgabe . Mathematische 94 open parenthesis Zeitschrift Jb , period B . t period 4 0 comma[ 41 ] 1\ 931quad38 comma ,L H . pp$ 5 ,\ period1ddot 934 ,{ pp4o 6} . closing 701$ - sung 708 parenthesis . der Aufgabe period Jahresberichte 94 ( Jb . der t . deutschen 4 0 , 1 Math 931 hyphen , pp . 4 6 ) . Jahresberichte der deutschen Math − ematikerematiker Vereinigung Vereinigung[ 40 ] Einige comma , weitere t period . 43 Konvergenzbedingungen 43 , comma vol . vol 1 − period4 der ,1933 1 Inversionss hyphen 4,pp. commaa¨ tze 4der 1 933− Limitierungsverfah5 comma . pp period - ren4 hyphen 5 period open square. bracketPublications 42 closing math e square´ matiques bracket de l ’ .. Universit Eine weiteree´ de UmkehrungBelgrade , t . des 2 , 1Ces 933 grave-a , pp . 1 roschen - 1 6 . Limitierungsverfahrens period[ 42 Glas] \quad Srpske[Eine 41 Kral ] hyphenweitereL o¨ sung derUmkehrung Aufgabe 94 des ( Jb . Ces t . 4 0 $ , 1\ 931grave , pp{ .a 4} 6$ ) . roschenJahresberichte Limitierungsverfahrens der deutschen . Glas Srpske Kral − jj evske evske Akademije AkademijeMath - ematiker comma , CLXIII CLXIII Vereinigung comma , Prvi , t . Prvi 43 razred , razredvol . 1- comma , 4 , 80 1 933 ,80, 1 comma pp 933 . 4 - 1 , 5 933 . pp comma . 59 − pp70 period . ( 59 Serbian hyphen 70 ) period . Bulletin open de parenthesisl ’ Acad Serbian $ [\ 42acute closing ] Eine{e parenthesis} weitere$ mie Umkehrung period royale Bulletin des serbe Ces de a` ,roschen S $ Limitierungsverfahrens\acute{e} $ rie . AGlas : SrpskeSciences Kral - math $ \acute{e} $ matiquesl quoteright ,j evske t Acad . 2Akademije acute-e , 1 935 mie , CLXIII royale, pp , Prvi. serbe 67 razred comma− 72 , 80 . S , e-acute1 933 , pp rie . A 59 : - Sciences 70 . ( Serbian math ) acute-e . Bulletin matiques de l ’ Acad comma t period 2 comma( German 1 935e summary´ commamie royale pp ) period serbe , 67 S hyphene´ rie A :72 Sciences period math e´ matiques , t . 2 , 1 935 , pp . 67 - 72 . ( German open parenthesissummary German ) summary closing parenthesis ¨ $open [ square 43 bracket ][43] \Uddot 43ber closing{ eineU} $ square berVe bracket et i− n er U-dieresiseilung\quad derVe ber Elemente eine$ t− ..r Ve $t-r komplexer eilung eilung der Doppelfolgen der Elemente Elemente . .. komplexer\quadRad Doppelfolgenkomplexer Doppelfolgen . \quad Rad \quad Jugoslavenske periodakademije .. RadJugoslavenske .. znanosti Jugoslavenske i akademije umjetnosti znanosti , i umjetnosti 2 5 1 , , Razred 2 5 1 , Razred matemati matemati $ cˇ\kocheck - prirodoslovni{c} $ ko, 78− , 1prirodoslovni , 78 , 1 935 , ppakademije . 223 −935 znanosti245 , pp . . i 223umjetnosti ( Serbian - 245 . ( comma Serbian ) . 2 )Bulletin 5 . 1Bulletin comma international internationalRazred matemati de l ’ Acad caron-c dee´ lmie ko ’ Acadyougoslave hyphen $prirodoslovni des\acute sciences{e} comma$ mie 78 comma yougoslave des sciences 1 935 comma et des beaux - arts , 28 , 1 934 , pp . 70 - 75 . ( German summary ) \ centerlinepp period 223{ et[ hyphen 44 des ] 245 beauxUn periodth −e´ or openartse` parenthesisme , 28 sur les , 1 Serbianint 934e´ grales closing, pp trigonom . parenthesis 70 − 75e´ triques period . (German . BulletinComptes international summary rendus du ) de} l quoteright e e` Acad acute-e2 miecongr yougoslaves des math des sciencese´ maticiens des pays slaves , 1 934 , Prague , 1 935 , pp . 147 - 149 . [ 44 ] \quad Un th $ \acute{e} $ or $ \grave{e} $ mesure´ les int $ \acute{e} $ grales trigonom et des beaux[ hyphen 45 ] Un arts apper commacedilla 28 comma− c u sur 1 934les inversions comma pp des periodproc 70d hyphene´ s de sommabilit 75 period opene´. Comptes parenthesis German e e` summary$ \acute closing{erendus} $ parenthesis du triques2 congr . Comptess des math renduse´ maticiens du des $ pays 2 slaves ˆ{ e ,} 1 934congr , Prague ˆ{\ , 1 935grave , pp{ .e 49}} - 61$ . s des mathopen square $ \acute bracket[ 46{ ]e} 44$Zu closing Fragen maticiens squareu¨ ber bracket des nichtve pays .. Unt slaves− thr acute-eauschbare , or 1 e-grave Grenzprozesse 934 , me Prague sur , les( ,mit int 1 acute-e H 935 . Wendelin , grales pp .) trigonom . 147 − e-acute149 . triques periodPublications Comptes rendus math due´ matiques 2 to the powerde l ’ Universit of e congre´ tode theBelgrade power , t of . 3grave-e , 1 934 s , des pp . 39 - 41 . ¨ [ 45math ] acute-e\quad maticiens[47]Un apperU ber des einige pays $ cedilla Inversionss slaves comma−c $a¨ 1tze 934 u der sur comma Limitierungsverfahren les Prague inversions comma . 1 935Publications des comma $ proc pp math period ˆ{\e´ matiques 147acute hyphen{e}} 149$ period d $ \acute{e} $ s deopen sommabilit squarede bracket l ’ Universit $ 45\ closingacutee´ de Belgrade{ squaree} { bracket , t. . 3}$ , 1.. 9 UnComptes 34 apper , pp . 1 cedilla-c rendus 53 - 1 60 u .du sur les $ inversions 2 ˆ{ e } des$ proc to the power of e-acute d acute-e$ congr s de sommabilit ˆ{\[ 48grave ] e-acuteQuelques{e}} sub$ th period se´ or des Comptese` mathmes de naturerendus $ \acute tauberienne du 2{ toe} the$ relatifs power maticiens aux of inte e´ desgrales pays et aux sslavese´ ries . , 1 934 , Prague , 1 935 , pp . 49 − 61 . congr to theGlas power Srpske of Kralj grave-e evske s des Akademije math e-acute , CLXV maticiens , Prvi razred des , pays8 1 , 1 slaves 935 , ppcomma . 1 73 1 - 934 2 29 comma . ( Serbian Prague ) . comma 1 935 comma[ 46 ] pp\quad periodBulletinZu 49 hyphen Fragen de l ’61 Acad period $ \e´ddotmie royale{u} $ serbe ber , S e nichtve´ rie A : Sciences $ t− mathr $e´ matiques auschbare , t . 2 Grenzprozesse , 1 935 , pp . , ( mit H . Wendelin ) . Publications mathopen square $ \1acute 69 bracket - 205{e} . 46 ($ Frenchclosing matiques summary square bracket de) l .. ’ Zu Universit Fragen u-dieresis $ \acute ber nichtve{e} $ t-r auschbare de Belgrade Grenzprozesse , t . comma3 , 1 open 934 , pp . 39 − 41 . e´ parenthesis mit H[ 49period ] WendelinRepr e´ sentation closing parenthesis des fonctions period doublement Publicationsp riodiques au moyen des int e´ grales d $math [ acute-e 47e´ fini ] matiques -\ esddot , de( avec{ lU quoteright} M$ . Petrovitch ber Universit einige ) . Glas e-acute Inversionss Srpske de Kraljevske Belgrade $ Akademijecomma\ddot t{ perioda ,} CLXV$ 3 comma,tze Prvi der razred 1 934 Limitierungsverfahren , 81comma , pp period 39 . Publications math hyphen$ \acute 41 period{e} $1 935 matiques , pp . 1 37 - 1 52 . ( Serbian ) . Bulletin de l ’ Acad e´ mie royale serbe , S e´ rie A : Sciences deopen l ’ square Universit bracket 47 $ closingmath\acutee´ squarematiques{e} $ bracket , det . 2 Belgrade U-dieresis, 1 935 , pp ber . , 239 t einige - . 243 3 Inversionss . , ( French 1934 summary a-dieresis , pp ) . tze 153 der− Limitierungsverfahren1 60 . ¨ period Publications[50] mathU ber acute-e einen matiques Konvergenzsatz des Herrn Knopp . Mathematische Zeitschrift , B . 40 , H . 3 , [ 48de l ] quoteright\quad1 935Quelques Universit , pp . 42 1 acute-e - th 425 . $ de\ Belgradeacute{e comma} $ or t period $ \ 3grave comma{e} 1 9$ 34 mescomma de pp nature period 1 tauberienne 53 hyphen 1 60 relatifsperiod aux int ¨ $ \openacute square{e} bracket$[ 5 1 grales ] 48Bemerkung closing et auxsquare zur s bracketNote $ \ ”acute ..U Quelquesber{e einige} $ th Inversionss acute-er i e s . or Glas e-gravea¨ tze der mes Limitierungsverfahren de nature tauberienne ” . relatifs aux int acute-eSrpske grales KraljPub et aux - evske lictions s e-acute Akademijemath riese´ periodmatiques , Glas CLXV de l ’ Universit , Prvie´ razredde Belgrade , 8 , t 1 . 4 , , 1 935 935 , pp , . pp 1 81 . - 1 1 84 73 . − 2 29 . ( Serbian ) . Srpske Kralj evske[ 52 ] AkademijeBemerkung commau¨ ber CLXV die vorstehende comma Prvi Arbeit razred des Herrncomma Avakumovi 8 1 commac´ 1, mit 935 n commaa¨ herer pp Betra period 1 73 hyphen 2Bulletin 29 period open de- chtungl parenthesis ’ Acad einer Klasse Serbian $ \acute von closing Funktionen{e} parenthesis$ welche mie royaleperiod bei den Inversionss serbe , Sa¨ tzen $ vorkommen\acute{e} . $ rie A : Sciences math $ \Bulletinacute{ dee} l$ quoterightRad matiques Jugoslavenske Acad , acute-e akademije t . 2 mie , znanosti royale 1 935 serbe i umjetnosti , comma , 254S e-acute , Razred rie matemati A : Sciencescˇ ko math - pri - acute-e rodoslovni matiques comma t periodpp . 2 1 comma 69,− 79 1 ,205 935 1 936 comma . , pp (French . 1 87 - 200 summary . ( Serbian ) ) . Bulletin international de l ’ Acad e´ mie pp period 1 69 hyphenyougoslave 205 des period sciences open et desparenthesis beaux - arts French , 2 9 summary , 1 936 , pp closing . 1 1 7 parenthesis - 1 23 . ( German summary ) ¨ [ 49open ] square\quad bracket[53]ReprU 49ber $ closing\ einigeacute square reihentheoretische{e} bracket$ sentation .. Repr S a¨ acute-etze des . Mathematische sentation fonctions des Zeitschriftfonctions doublement doublement , B . 41 , $ H p . p 1 ˆ to,{\ 1 the 936acute power , { ofe}} e-acute$ riodiques au moyen des int riodiques$ \acute au{e moyenpp} $ . 67 des g- r74 a int l. e sacute-e d $ grales\acute d e-acute{e} $ fini f hyphen i n i − ¨ eses ,comma ( avec open M[54] parenthesis . PetrovitchU ber einige avec Taubersche M ) period . Glas Petrovitch S a¨ Srpsketze deren closing Kraljevske Asymptotik parenthesis von Akademije periodExponentialcharakter Glas Srpske , CLXV Kraljevske ist ,. I Prvi , ( mit Akademije razred comma , 81 , CLXV comma Prvi razredV . Avakumovi comma 81c´). commaMathematische Zeitschrift , B . 41 , H . 3 , 1 936 , pp . 345 - 356 . \ hspace1 935 comma∗{\ f i lpp l } period1 935 1 37 , hyphenpp . 1 1 5237 period− 1 open52 . parenthesis ( Serbian Serbian ) . closing Bulletin parenthesis de l period ’ Acad Bulletin $ \ deacute l quoteright{e} $ Acadmie royaleacute-e mie serbe royale , serbe S $ comma\acute S e-acute{e} $ rie rie A : Sciences A : Sciences math acute-e matiques comma t period 2 comma 1 935 comma pp period 239 hyphen 243 period open parenthesis French summary\ centerline closing{math parenthesis $ \acute{e} $ matiques , t . 2 , 1 935 , pp . 239 − 243 . ( French summary ) } open square bracket 50 closing square bracket U-dieresis ber einen Konvergenzsatz des Herrn Knopp period Mathematische Zeitschrift$ [ 50 comma ] B period\ddot 40{U comma} $ ber H period einen Konvergenzsatz des Herrn Knopp . Mathematische Zeitschrift , B . 40 , H . 33 ,1935 comma 1 935 ,pp comma . 421 pp period− 425 42 . 1 hyphen 425 period open square bracket 5 1 closing square bracket .. Bemerkung zur Note quotedblright U-dieresis ber einige Inversionss a-dieresis[ 5 1 ] tze\quad der LimitierungsverfahrenBemerkung zur Note quotedblright ’’ $ \ periodddot{U Pub} $ hyphen ber einige Inversionss $ \ddot{a} $ tze der Limitierungsverfahren ’’ . Pub − lictionslictions math math acute-e $ \ matiquesacute{e de} l$ quoteright matiques Universit de l e-acute ’ Universit de Belgrade $ comma\acute t{ periode} $ 4 decomma Belgrade 1 935 comma , t .pp 4 , 1 935 , pp . 1 81 − 1 84 . period 1 81 hyphen 1 84 period [ 52open ] square\quad bracketBemerkung 52 closing $ square\ddot bracket{u} $ .. ber Bemerkung die vorstehende u-dieresis ber die Arbeit vorstehende des Herrn Arbeit Avakumovides Herrn Avakumovi $ \acute{c} { , }$ c-acutemit n sub $ comma\ddot{ mita} n$ dieresis-a herer herer Betra Betra− hyphen chtungchtung einer einer Klasse Klasse von Funktionen von Funktionen welche bei welche den Inversionss bei den dieresis-a Inversionss tzen vorkommen $ \ddot period{a} $ tzen vorkommen . Rad Jugoslavenske akademije znanosti i umjetnosti comma 254 comma Razred matemati caron-c ko hyphen pri hyphen Radrodoslovni Jugoslavenske comma 79 akademije comma 1 936znanosti comma pp period i umjetnosti 1 87 hyphen , 200 254 period , Razred open parenthesis matemati Serbian $ \ closingcheck parenthesis{c} $ ko − p r i − periodrodoslovni Bulletin international , 79 , 1 936 de l quoteright , pp . 1 Acad 87 − acute-e200 mie . ( Serbian ) . Bulletin international de l ’ Acad $ \yougoslaveacute{e} des$ sciences mie et des beaux hyphen arts comma 2 9 comma 1 936 comma pp period 1 1 7 hyphen 1 23 period open parenthesis German summary closing parenthesis \ hspaceopen square∗{\ f i bracket l l } yougoslave 53 closing square des bracket sciences U-dieresis et des ber einige beaux reihentheoretische− arts ,29 S ,1936a-dieresis tze ,pp period . 117Mathematische− 1 23 . ( German summary ) Zeitschrift comma B period 41 comma H period 1 comma 1 936 comma $pp [ period 53 67 ]hyphen\ddot 74 period{U} $ ber einige reihentheoretische S $ \ddot{a} $ tze . Mathematische Zeitschrift , B . 41 , H . 1 , 1 936 , ppopen . 67 square− 74 bracket . 54 closing square bracket U-dieresis ber einige Taubersche S a-dieresis tze deren Asymptotik von Exponentialcharakter ist period I comma open parenthesis mit \ hspaceV period∗{\ Avakumovif i l l } $ acute-c[ 54 closing ] parenthesis\ddot{U period} $ berMathematische einige Taubersche Zeitschrift comma S $ B\ periodddot{ 41a} comma$ tze H period deren 3 Asymptotik von Exponentialcharakter ist . I , ( mit comma 1 936 comma pp period 345 hyphen 356 period \ centerline {V . Avakumovi $ \acute{c} ) . $ Mathematische Zeitschrift , B . 41 , H . 3 , 1 936 , pp . 345 − 356 . } Jovan Karamata .... 25 \noindentopen squareJovan bracket Karamata 55 closing\ squareh f i l l bracket25 .. A llgemeine Umkehrs dieresis-a tze der Limitierungsverfahren period Abhandlungen aus dem Math hyphen [ 55ematischen ] \quad SeminarA llgemeine der Hansischen Umkehrs Universit $ a-dieresis\ddot{a} t comma$ tze t period der Limitierungsverfahren 1 2 comma 1 comma 1 937 comma . Abhandlungen pp period 48 aus dem Math − hyphenematischen 63 period Seminar der Hansischen Universit $ \ddot{a} $ t ,t.12,1,1937,pp.48 − 63 . open square bracket 56 closing square bracket U-dieresis ber allgemeine Umkehrs a-dieresis tze der Limitierungsverfahren period$ [ Comptes 56 rendus] \ddot du Con{U} hyphen$ ber allgemeine Umkehrs $ \ddot{a} $ tze der Limitierungsverfahren . Comptes rendus du Con − $gr gr to the ˆ{\ powerJovangrave Karamataof{ grave-ee}}$ s international s international des math des e-acute math maticiens $ \acute comma{e 1} 936$ comma maticiens Oslo comma , 125 936 II comma , Oslo 1 937 , II , 1 937 , pp . 1 1 6 − 1 1 7 . comma pp period[1 55 1 ] 6 hyphenA llgemeine 1 1 7 Umkehrs period a¨ tze der Limitierungsverfahren . Abhandlungen aus dem Math - $open [ square 57ematischen bracket ] \ddot Seminar57 closing{U} der$ square Hansischen ber bracket einen Universit U-dieresis Satza¨ vont , bert . H 1 einen 2 . , 1 Heilbronn , Satz 1 937 von , pp H . und 48 period - 63 E . Heilbronn. Landau und . Publications E period Landau math ¨ e` period$ \acute Publications{e} $[56] math matiquesU ber e-acute allgemeine de matiques Umkehrs de a¨ tze der Limitierungsverfahren . Comptes rendus du Con - gr ll ’ quoteright Universits international Universit $ \ acute-eacute des math{ dee} Belgradee´$maticiens de comma Belgrade , 1 936 t period, Oslo , t, 5 II comma. , 1 5 937 , ,1 pp 936 936 . 1 comma 1 , 6 -pp 1 pp 1 . 7 period . 28 − 2838 hyphen . 38 period ¨ open square bracket[57] U 58ber closing einen square Satz von bracket H . Heilbronn .. Compl und acute-e E . Landau ments . auPublications th e-acute math or grave-ee´ matiques me d quoteright de l ’ Abel et de son[ 58 inverse ] \quad periodUniversitCompl Revuee´ de math $ Belgrade\acute e-acute , t{ matique .e} 5 ,$ 1 936 ments de , ppl quoteright . 28 au - 38 th . Union $ \acute{e} $ or $ \grave{e} $ me d ’ Abel et de son inverse . Revue math $ \interbalkaniqueacute{e} $[ 58 comma matique ] Compl 1 open dee´ parenthesisments l ’ Union au th 2 closinge´ or e` parenthesisme d ’ Abel comma et de son 1 936inverse comma . Revue pp period math 1e´ 61matique hyphen 1 65 period interbalkaniqueopen squarede bracket l ’ Union , 59 interbalkanique1 closing ( 2 square) , 1 , bracket 1 936 ( 2 ) , , .. 1 pp Quelques936 . , pp 1 . 61 1moyens 61− -1 1 65 pa 65 . to . the power of r-t iculiers pour e-acute tablir des th r−t e´ acute-e or e-grave[ 59mes ] inversesQuelques des moyens proc to thepa powericuliers of acute-e pour e´ dtablir e-acute des s th de e´ or e` mes inverses des proc [ 59 ] \quad Quelques moyens $ pa ˆ{ r−t }$ iculiers pour $ \acute{e} $ tablir des th $ \acute{e} $ sommabilitd acute-ee´ s de sub sommabilit period Revistae´. Revista de Ciencias de Ciencias comma , Universidad Mayor Mayor de de San San Marcos Marcos , Lima comma , Facultad Lima comma Facultad orde $ Ciencias\gravede Biologicas Ciencias{e} $ Biologicas mes comma inverses , Fisicas Fisicas y y des Matem $a´ acute-a procticas , ˆ tticas{\ . 38acute comma, vol .{ 418e t}}period , 1$ 936 d , 38 pp comma $ . 1\ 5acute 5 - vol 1 60{ periode .} $ 418 s decomma 1 936 sommabilit $ \acute{e} { . }$ Revista de Cienciase´ , Universidad Mayor de San Marcos , Lima , Facultad comma pp period[ 1 60 5 ] 5 hyphenSur les 1 th 60 periode´ or e` mes inverses des proc d e´ s de sommabilit e´. Actualit e´ s deopen Ciencias squarescientifiques bracket Biologicas 60 et closing industrielles , square Fisicas , bracket 450 , y Hermann Matem.. Sur les et Cie $th\ acute-e ,acute Paris , or{ 1a 9 e-grave} 37$ , p .ticas mes 47 . inverses , t des . 38 proc , to vol the .power 418 of ,1936 acute-e ,pp . 155 − 1 60 . d e-acute s de sommabilit[ 61 acute-e] Beziehungen sub period zwischen Actualit der e-acute Oszillationsgrenzen s scientifiques einer et Funktion und ihrer arithmetis - [ 60industrielles ] \quad commachenSur Mittel l 450 e s comma. thProceedings $ Hermann\acute of the{ ete} CieLondon$ comma or Mathematical$ Paris\grave comma Society{e} 1 9$ , 37 mes comma 2 ( 43 inverses ) p, period 1 937 47 des , pp period . $ 20 proc - ˆ{\acute{e}}$ dopen $ \acute square25{bracket .e} $ 61 s declosing sommabilit square bracket $ ..\acute Beziehungen{e} { zwischen. }$ der A c Oszillationsgrenzen t u a l i t $ \acute einer{e} Funktion$ s scientifiques und ihrer et industrielles , 450 , Hermann et Cie , Paris , 1 9 37 , p . 47R . 1/σ arithmetis hyphen[ 62 ] Un th e´ or e` me relatif aux sommabilit e´ s de la forme σ 0 ψ(σt)s(t)dt. V e˘ chen Mittelstnik period Kr a Proceedings´ l . Cˇ esk e of´ spole the Londoncˇ nosti naukMathematical , II , 1 936 Society , XIX , comma1 937 , p .. . 25 .open parenthesis 43 closing parenthesis comma\ hspace ..∗{\ 1 937f i comma l[ l 63} [ ] 61 ppEin period ] Konvergensatz\quad Beziehungen f u¨ r trigonometrische zwischen Integrale der Oszillationsgrenzen . Journal f u¨ r die reine und einer ange Funktion - und ihrer arithmetis − 20 hyphenwandte 25 period Mathematik , 1 78 , 1 937 , pp . 29 - 33 . chenopen Mittel square bracket .[ 64 Proceedings ] 62Einige closing square of S thea¨ brackettze London ..u¨ Unber Mathematical th acute-e die or Rieszschen e-grave Society me relatif Mittel , \quad aux . sommabilit2Glas ( 43 Srpske acute-e ) , \quad s de la1 forme 937 , pp . sigma20 − integral25 . Kralj sub 0 evske to the power Akademije of 1 slash , CLXXV sigma , psiPrvi open razred parenthesis , 86 , pp sigma . 2 9 t 1 closing - 326 . parenthesis ( Serbian s )open . Bulletin parenthesis t closing parenthesis dtde period l ’ Acad V breve-ee´ mie royale stnik Kr acute-a l period [ 62C-caron ] \quad esk e-acuteserbeUn th , Sspolee´ $rie caron-c\ Aacute : Sciences nosti{e} math nauk$ ore´ commamatiques $ \ IIgrave comma, t . 4{ ,e 1 1} 938 936$ , commapp me . 1 relatif 2 XIX1 - 1 37 comma . aux ( German 1 sommabilit 937 summary comma p ) period $ \acute 5 period{e} $ s deopen la square forme bracket[65] $ \ 63sigmaU¨ closingber allgemeine square\ int bracketˆ O{ - Umkehrs1 .. Ein / Konvergensatza¨\sigmatze . Rad} Jugoslavenske{ f u-dieresis0 }\ psi r akademije trigonometrische( znanosti\sigma Integralei umjett - period ) Journal s ( ft dieresis-u ) dt r die reinenosti . $ ,und 261 V ange , Razred$ \ hyphenbreve matemati{e} $cˇ ko s -t nprirodoslovni i k Kr $ ,\ 81acute , 1 938{a ,} pp$ . 1 - l 2 .2 . ( Serbian ) . Bulletin $wandte\check Mathematikinternational{C} $ esk comma de l$ ’ Acad 1\ 78acute commae´ mie{e yougoslave} 1$ 937 scomma p o des l e sciences pp $ period\check et des 29{ beaux hyphenc} $ - arts 33 nosti period , 32 , nauk 1 939 , ,pp II . 1 , - 91 . 936 ( ,XIX , 1 937 , p . 5 . open squareGerman bracket summary 64 closing ) square bracket .. Einige .. S dieresis-a tze .. dieresis-u ber .. die .. Rieszschen .. Mittel period[ 63 ..] Glas\quad .. Srpske[66]EinU¨ Konvergensatz ..ber Kralj die evskeB - Limitierbarkeit .. Akademije f $ \ einerddot comma Potenzreihe{u} $ r am trigonometrische Rande . Mathematische Integrale Zeitschrift , . B Journal . f $ \ddot{u} $ r dieCLXXV reine comma44 ,und H . Prvi 1 ange , 1 razred 938− , pp comma . 1 56 ..- 1 86 60 comma . pp period 2 9 1 hyphen 326 period .. open parenthesis Serbian closing parenthesiswandte Mathematik period[ 67 Bulletin ] Un , de 1 th l 78 quoterighte´ or , 1e` 937me Acad sur , le acute-e ppproc . 29 miee´ d− royalee´33de sommabilit . e´ de Borel . Publications math serbe commae´ matiques S acute-e de rie l ’ Universit A : Sciencese´ de math Belgrade e-acute , t . matiques 6 - 7 , 1 937 comma - 38 ,pp t period . 204 - 4 2 comma 8 . 1 938 comma pp period 1 2 1 hyphen[ 64 ] 1\ 37quad period[Einige 68 open ] parenthesisQuelques\quad thS Germane´ $ or\ddot summarye` mes{a} sur$ closingles t int z e parenthesis\e´quadgrales de$ Laplace\ddot -{u A} bel$ . berBulletin\quad mathde i´ e \quad Rieszschen \quad Mi tte l . \quad Glas \quad Srpske \quad Kralj evske \quad Akademije , CLXXVopen square , Prvimatique bracket razred de la 65 Soci closing , \quade´ t squaree´ roumaine86 bracket , pp des . U-dieresisSciences 2 9 1 ,− 40 ber (326 1 allgemeine - 2 . ) ,\ 1quad 938 O , pp( hyphen . Serbian 1 7 - Umkehrs 1 8 . ) . a-dieresis Bulletin tze de period l ’ Rad Acad Jugoslavenske$ \acute{e} akademije$[ 69 mie ] znanostiSur r o y la a sommabilitl e i umjet hyphenefo´ t−r e et la sommabilit e´ absolue . Mathematica ( Cluj ) , 1 5 , 1 nosti comma9 39 261 , pp comma . 1 1 9 Razred - 1 24 . matemati caron-c ko hyphen prirodoslovni comma 81 comma 1 938 comma pp period 1 hyphen\ hspace 2∗{\ 2 periodf i l[ l 70 open} serbe ] parenthesisEinige , S S a¨ $ Serbiantze\acuteu¨ ber closing{ iteriertee} parenthesis$ Mittelbildungen rie A period : Sciences .BulletinProceedings math $ of\ theacute The{ Benarese} $ Ma matiques , t . 4 , 1938 , pp . 121 − 1 37 . ( German summary ) international- thematical de l quoteright Society Acad ( New acute-e Series ) mie , 1 , yougoslave 1 939 , pp . des 1 5 sciences - 24 . et des beaux hyphen arts comma 32 comma 1 939 comma\ hspace pp∗{\ periodf i l[71] l } $U¨ [ber einen 65 Tauberschen ] \ddot Satz{U im} Dreik$ bero¨ rperproblem allgemeine . American O − Umkehrs Journal of Mathe $ \ddot - matics{a} $ tze . Rad Jugoslavenske akademije znanosti i umjet − 1 hyphen 9, vol period . 61 open , 3 , 1 parenthesis 939 , pp . 769 German - 770 . summary closing parenthesis nostiopen , square 261 bracket ,[ 72 Razred ] 66On closing infinitelymatemati square increasinging $bracket\check sequences U-dieresis{c} $ . berMatemati ko die− Bprirodoslovni hyphencˇ ki Vesnik Limitierbarkeit ( Assoc , . 81students einer, 1 of 938 Potenzreihe math , . pp . am 1 Rande− 2 2 . ( Serbian ) . Bulletin periodinternational Mathematische de Zeitschrift l ’ Acad commaUniv $ . Belgrade\acute{ )e 5} - 6$ , 1 939mie , pp yougoslave . 1 1 - 36 . ( Serbiandes sciences ) et des beaux − arts , 32 , 1 939 , pp . 1 B− period9 . ( 44 German comma H summary period 1 comma ) 1 938 comma pp period 1 56 hyphen 1 60 period open square bracket 67 closing square bracket .. Un th acute-e or e-grave me sur le proc to the power of acute-e d e-acute de$ sommabilit [ 66 acute-e ] \ deddot Borel{U} period$ ber Publications d i e B − mathLimitierbarkeit e-acute matiques einer Potenzreihe am Rande . Mathematische Zeitschrift , B.de l 44 quoteright ,H. Universit 1 ,1938 acute-e ,pp. de Belgrade 156 − comma1 60 t period . 6 hyphen 7 comma 1 937 hyphen 38 comma pp period 204 hyphen 2 8 period [ 67open ] square\quad bracketUn th 68 closing$ \acute square{e} bracket$ or .. Quelques $ \grave th{ acute-ee} $ or mesur e-grave le mes sur $ proc les int ˆ acute-e{\acute grales{e}} de Laplace$ d hyphen$ \acute A bel{e} period$ de Bulletin sommabilit math e-acute $ matique\acute{ dee} la$ de Borel . Publications math $ \acute{e} $ matiques deSoci l acute-e’ Universit t e-acute $ roumaine\acute des{e} Sciences$ de comma Belgrade 40 open , t parenthesis . 6 − 7 1 , hyphen 1 937 2− closing38 , parenthesis pp . 204 comma− 2 8 1 . 938 comma pp period 1 7 hyphen 1 8 period [ 68open ] square\quad bracketQuelques 69 closing th square $ \acute bracket{e} .. Sur$ la or sommabilit $ \grave acute-e{e} fo$ to mes the power sur ofles t-r int e et la $ sommabilit\acute{ acute-ee} $ absoluegrales period de Laplace Mathematica− A open bel parenthesis . Bulletin Cluj math closing $parenthesis\acute{ commae} $ 1 matique 5 comma 1de 9 39 la comma Socipp period $ \acute 1 1 9 hyphen{e} $ 1 24 t period $ \acute{e} $ roumaine des Sciences , 40 ( 1 − 2) ,1938 ,pp.17 − 1 8 . open square bracket 70 closing square bracket .. Einige S dieresis-a tze dieresis-u ber iterierte Mittelbildungen period .. Proceedings[ 69 ] \quad of theSur The la Benares sommabilit Ma hyphen $ \acute{e} f o ˆ{ t−r }$ e et la sommabilit $ \acute{e} $ absoluethematical . Mathematica Society open parenthesis ( Cluj ) New , 1Series 5 , closing 1 9 39 parenthesis , comma 1 comma 1 939 comma pp period 1 5 hyphen 24pp period . 1 1 9 − 1 24 . open square bracket 71 closing square bracket U-dieresis ber einen Tauberschen Satz im Dreik dieresis-o rperproblem period American[ 70 ] \ Journalquad Einige of Mathe S hyphen $ \ddot{a} $ t z e $ \ddot{u} $ ber iterierte Mittelbildungen . \quad Proceedings of the The Benares Ma − thematicalmatics comma Society vol period ( 61 New comma Series 3 comma ) , 1 1 939 ,comma 1 939 pp , ppperiod . 1 769 5 hyphen− 24 770. period open square bracket 72 closing square bracket .. On infinitely increasinging sequences period Matemati caron-c ki Vesnik open$ [ parenthesis 71 ] Assoc\ddot period{U students} $ ber of math einen period Tauberschen Satz im Dreik $ \ddot{o} $ rperproblem . American Journal of Mathe − maticsUniv period , vol Belgrade . 61 closing , 3 , parenthesis 1 939 , 5 pp hyphen . 769 6 comma− 770 1 939 . comma pp period 1 1 hyphen 36 period open parenthesis Serbian closing parenthesis \ hspace ∗{\ f i l l } [ 72 ] \quad On infinitely increasinging sequences . Matemati $ \check{c} $ ki Vesnik ( Assoc . students of math .

\ centerline {Univ . Belgrade ) 5 − 6 ,1939 ,pp. 11 − 36 . ( Serbian ) } 26 M . Tomi c´ [ 73 ] Sur le th e´ or e` me tauberien de N . Wiener . Publications de l ’ Institut math e´ matique de l ’ Acad e´ mie serbe , 3 , 1 950 , pp . 20 1 - 206 . ( written in 1 939 ) [74] U¨ ber die Indexverschiebung beim Borelschen Limitierungsverfahren . Mathematische Zeitschrift , B . 45 , H . 5 , 1 939 , pp . 635 - 641 . [ 75 ] Quelques th e´ or e` mes inverses relatifs aux proce´ d e´ s de sommabilit e´ de Ces a` ro et Riesz . Glas Srpske Akademije Nauka , CXCI , Prvi razred , 96 , 1 949 , pp . 1 - 37 . ( Serbian ) . Publications de l ’ Institut math e´ matique de l ’ Acad e´ mie serbe , 3 , 1 9 50 , pp . 53 - 71 . ( French summary ) [ 76 ] A note on convergence factors . Journal of the London Mathematical Society , 2 1 , 1 946 , pp . 1 62 - 1 66 . 0 [ 77 ] Sur l ’ application des th e´ or e` mes de nature tauberienne a` l e´ tude des valeurs asymp - totiques des e´ quations diff e´ rentielles . Publications de l ’ Institut math e´ matique de l ’ Acad e´ mie serbe , 1 , 1 947 , pp . 93 - 9 6 . [ 78 ] Sur le sommabilit e´ de S . Bernstein et quelques proce´ d e´ s de sommation qui s ’ y rat - tachent . Matemati cˇ eskii Sbornik , Moscow , n . s . 2 1 ( 63 ) , No . 1 , 1 947 , pp . 1 3 - 24 . [ 79 ] Sur certains sommes trigonom e´ triques en rappot−r aux approximations diophanti - ennes . Comptes rendus du 63e Congr e` s de l ’ A . F . A . S . , Biarritz , 1 947 . ( Russian sum -

mary) [ 80 ] In e´ galit e´ s relatives aux quotients et a` la diff e´ rence de R fg et R f R g. Publications de l ’ Institut math e´ matique de l ’ Acad e´ mie serbe , 2 , 1 948 , pp . 1 31 - 145 . [ 81 ] Consid e´ rations ge´ om e´ triques relatives aux polyn oˆ mes et s e´ ries trigonom e´ triques , ( avec M . Tomi c´). Publications de l ’ Instinstitut math e´ matique de l ’ Acad e´ mie serbe , 2 , 1 948 , pp . 1 57 - 1 75 . ( Serbian summary ) [ 82 ] Sur une in e´ galit e´ de Kusmin - Landau relative aux sommes trigonom e´ triques et son application a` la somme de Gauss , ( avec M . Tomi c´). Glas Srpske Akademije Nauka , CXCVIII , n . s . 3 , 1 9 50 , pp . 1 63 - 1 74 . ( Serbian ) . Publications de l ’ Institut math e´ matique de l ’ Acad e´ mie serbe , 3 , 1 950 , pp . 207 - 2 1 8 . ( French ) [ 83 ] Sur l ’ approximation de ex par des fonctions rationneles . Bulletin de la Soci e´ t e´ des math e´ maticiens et physiciens de Serbie ( Vesnik ) , I - 1 , 1 949 , pp . 7 - 1 9 . ( Serbian ; Rus - sian and French summaries ) [ 84 ] Mihailo Petrovi c´, 24.4.1868 − 8.6.1943. ( in memoriam ) Glasnik matemati cˇ ko - fizi cˇ ki i astronomski , ser . II , t . 3 , vol . 3 , 1 948 , pp . 1 23 - 1 2 7 . ( Serbian ) [85] U¨ ber die Beziehung zwischen dem Bernsteinschen und Ces a` roschen Limitierungsver - fahren . Mathematische Zeitschrift , B . 52 , H . 3 , 1 949 , pp . 305 - 306 . [ 86 ] Appendix to the paper of J . Hlit cˇ ijev : On longitudinal ribs in a compressed place . Glas Srpske Akademije Nauka , CXCV , Odeljenje tehni cˇ kih nauka , 1 , 1 949 , pp . 1 7 - 36 . ( Serbian ) [ 87 ] A Survey of Elementary Mathematics . Pa r − t I : Arithmetic and Algebra . Nau cˇ na kn - j iga , Belgrade , 1 949 , p . 63 . ( Serbian ) [ 88 ] Theory and Practice of the Stieltjes Integral . Srpska Akademij a Nauka , Posebna izdanj a CLIV , Matemati cˇ ki institut 1 , Belgrade , 1 949 , p . 328 . ( Serbian ) 26 .... M period Tomi acute-c \noindentopen square26 bracket\ h f i l 73 l M closing . Tomi square $ bracket\acute ..{ Surc} le$ th acute-e or e-grave me tauberien de N period .. Wiener period Publications de l quoteright Institut math acute-e matique [ 73de l ] quoteright\quad Sur Acad l acute-ee th mie $ \ serbeacute comma{e} $ 3 comma or $ 1\ 950grave comma{e} pp$ period me tauberien 20 1 hyphen 206 de periodN . \quad open parenthesisWiener . Publications de l ’ Institut math written$ \acute in 1{e 939} $ closing matique parenthesis deopen l ’square Acad bracket $ \acute 74 closing{e} square$ mie bracket serbe U-dieresis , 3 ,ber 1 die 950 Indexverschiebung , pp . 20 1 beim− 206 Borelschen . ( written Limitierungsverfahren in 1 939 ) period Mathematische $Zeitschrift [ 74 comma ][ 89 ]\ Bddot periodComplex{U} 45$ Numbers comma ber H with die period Applications Indexverschiebung 5 comma to 1 Elementary 939 comma Geometrybeim pp period Borelschen . 635Nau hyphencˇ na Limitierungsverfahren knj 641 iga period , Bel - . Mathematische Zeitschriftopen squaregrade bracket , , B 1 950 . 75 45, closing p . ,H 1 60 square . . (5 Serbian bracket, 1 ) 939 .. Quelques , pp . th 635 acute-e− 641 or e-grave . mes inverses relatifs aux proc to the power of acute-e d e-acute s de sommabilit acute-e de Ces a-grave ro et Riesz period [ 75Glas ] Srpske\quad AkademijeQuelques Nauka th comma $ \acute CXCI{e comma} $ or Prvi $razred\grave comma{e} 96$ comma mes 1inverses 949 comma relatifs pp period aux1 hyphen $ proc 37 ˆ{\acute{e}}$ periodd $ \ openacute parenthesis{e} $ Serbian s de sommabilit closing parenthesis $ \ periodacute{e} $ de Ces $ \grave{a} $ ro et Riesz . GlasPublications Srpske de Akademije l quoteright Nauka Institut , math CXCI acute-e , Prvi matique razred de l quoteright , 96 , 1 Acad 949 e-acute , pp mie. 1 serbe− 37 comma . ( Serbian 3 comma 1 ) 9 . 50 comma pp period 53 hyphen 71 period Publicationsopen parenthesis de French l ’ Institut summary closing math parenthesis $ \acute{e} $ matique de l ’ Acad $ \acute{e} $ mie serbe , 3 , 1950 , pp . 53 − 71 . (open French square summary bracket 76 ) closing square bracket .. A note on convergence factors period Journal of the London Mathematical Society comma 2 1 comma [ 761 946 ] comma\quad ppA periodnote on 1 62 convergence hyphen 1 66 period factors . Journal of the London Mathematical Society , 2 1 , 1open 946 square , pp bracket. 1 62 77− closing1 66 square. bracket .. Sur l quoteright application des th acute-e or e-grave mes de nature tauberienne grave-a l quoteright sub e-acute tude des valeurs asymp hyphen [ 77totiques ] \quad des acute-eSur l quations ’ application diff e-acute rentielles des th period $ \acute Publications{e} $ de or l quoteright $ \grave Institut{e} math$ mes acute-e de matiquenature de tauberienne $ \lgrave quoteright{a} Acad$ l acute-e $ ’ mie{\ serbeacute comma{e}} 1$ comma tude 1 947 des comma valeurs pp period asymp 93− hyphen 9 6 period totiquesopen square des bracket $ \ 78acute closing{e} square$ quations bracket .. Sur diff le sommabilit $ \acute acute-e{e} $ de S rentielles period Bernstein . Publications et quelques proc de to the l ’ Institut math power$ \acute of e-acute{e} $ d acute-e matique s de desommation qui s quoteright y rat hyphen ltachent ’ Acad period $ \ Matematiacute{e} caron-c$ mie eskii serbe Sbornik , comma 1 , 1 Moscow 947 , comma pp . 93n period− 9 s 6 period . 2 1 open parenthesis 63 closing parenthesis comma No period 1 comma 1 947 comma pp period 1 3 hyphen 24 period \ hspaceopen square∗{\ f i bracket l l } [ 78 79 closing ] \quad squareSur bracket le sommabilit .. Sur certains $ sommes\acute trigonom{e} $ acute-e de S triques. Bernstein en rappo et to the quelques power of t-r$ proc aux approximations ˆ{\acute{e diophanti}}$ d hyphen $ \acute{e} $ s de sommation qui s ’ y rat − ennes period Comptes rendus du 63 to the power of e Congr grave-e s de l quoteright A period F period A period S period comma\ centerline Biarritz{ commatachent 1 947 . Matematiperiod open parenthesis $ \check Russian{c} $ sum eskii hyphen Sbornik , Moscow , n . s . 21 ( 63 ) ,No . 1 , 1 947 , pp . 13 − 24 . } mary closing parenthesis [ 79open ] square\quad bracketSur certains 80 closing square sommes bracket trigonom .. In acute-e $ \ galitacute e-acute{e} $ s relatives triques aux en quotients $ rappo et grave-a ˆ{ t la− diffr } e-acute$ aux approximations diophanti − renceennes de integral . Comptes fg et integral rendus f integral du $ g 63 period ˆ{ e Publications}$ Congr $ \grave{e} $ s de l ’A. F .A. S . , Biarritz , 1 947 . ( Russian sum − de l quoteright Institut math acute-e matique de l quoteright Acad e-acute mie serbe comma 2 comma 1 948 comma pp period\ [ mary 1 31 hyphen ) \ ] 145 period open square bracket 81 closing square bracket .. Consid acute-e rations g to the power of e-acute om acute-e triques relatives aux polyn circumflex-o mes .. et s acute-e ries trigonom e-acute triques comma \ hspaceopen parenthesis∗{\ f i l l } [ avec 80 M ] period\quad TomiIn acute-c $ \acute closing{e} parenthesis$ g a l i t period $ \ Publicationsacute{e} $ de ls quoteright relatives Instinstitut aux quotients math et e-acute$ \grave matique{a} $ de l l quoteright a d i f f Acad $ \acute acute-e{e mie} $ serbe rence comma de $ \ int f g $ et $ \ int f \ int g . $2 comma Publications 1 948 comma pp period 1 57 hyphen 1 75 period open parenthesis Serbian summary closing parenthesis open square bracket 82 closing square bracket .. Sur une .. in acute-e galit e-acute .. de .. Kusmin hyphen Landau .. relative\ centerline .. aux sommes{de l .. ’ trigonom Institut acute-e math triques $ \ ..acute et {e} $ matique de l ’ Acad $ \acute{e} $ mie serbe , 2 , 1 948 , pp . 1 31 − 145 . } son application grave-a la somme de .. Gauss comma .. open parenthesis avec M period Tomi c-acute closing parenthesis period[ 81 ..] Glas\quad SrpskeConsid Akademije $ \acute{e} $ rations $gˆ{\acute{e}}$ om $ \acute{e} $ triques relatives aux polyn $ \Naukahat{o comma} $ mes CXCVIII\quad commaet s n period $ \acute s period{e} 3$ comma ries 1 9 trigonom 50 comma pp $ period\acute ..{ 1e 63} $ hyphen t r i q 1 u 74 e s period , .. open parenthesis( avecM Serbian . Tomi closing $ \ parenthesisacute{c} period) Publications . $ Publications de l quoteright de Institut l ’ Instinstitut math $ \acute{e} $ matiquemath acute-e de l matique ’ Acad de $ l quoteright\acute{e Acad} $ e-acute mie serbe mie serbe , comma 3 comma 1 950 comma pp period 207 hyphen 2 1 8 period open parenthesis French closing parenthesis \ centerlineopen square{ bracket2 ,1948 83 closing ,pp square . 157 bracket− 1 .. 75 Sur . l quoteright ( Serbian approximation summary ) de} e to the power of x par des fonctions rationneles period Bulletin de la Soci acute-e t e-acute des [ 82math ] acute-e\quad maticiensSur une et\quad physiciensin de $ Serbie\acute open{e} parenthesis$ g a l i t Vesnik $ \ closingacute{ parenthesise} $ \quad commade I\ hyphenquad 1Kusmin comma− 1 Landau \quad r e l a t i v e \quad aux sommes \quad trigonom 949$ \ commaacute{ ppe} period$ t r 7 i qhyphen u e s \ 1quad 9 periodet open parenthesis Serbian semicolon Rus hyphen sonsian application and French summaries $ \grave closing{a} parenthesis$ l a somme de \quad Gauss , \quad ( avecM . Tomi $ \acute{c} )open . $ square\quad bracketGlas 84 closing Srpske square Akademije bracket .. Mihailo .. Petrovi acute-c sub comma 24 period 4 period 1 868 hyphen 8 periodNauka 6 period , CXCVIII 1 943 period , n .open s .parenthesis 3 , 1 9 in 50 .. memoriam , pp . \ closingquad 1 parenthesis 63 − 1 74.. Glasnik . \quad .. matemati( Serbian c-caron ) ko .hyphen Publications de l ’ Institut fizi caron-c ki i astronomski comma ser period II comma t period 3 comma vol period 3 comma 1 948 comma pp period 1 23\ centerline hyphen 1 2 7{math period open $ \acute parenthesis{e} $ Serbian matique closing de parenthesis l ’ Acad $ \acute{e} $ mie serbe , 3 , 1 950 , pp . 207 − 2 1 8 . ( French ) } open square bracket 85 closing square bracket U-dieresis ber die Beziehung zwischen dem Bernsteinschen und Ces a-grave roschen[ 83 ] Limitierungsver\quad Sur l hyphen ’ approximation de $ e ˆ{ x }$ par des fonctions rationneles . Bulletin de la Soci $ \fahrenacute period{e} $ .. Mathematische t $ \acute{ Zeitschrifte} $ des comma B period 52 comma H period 3 comma 1 949 comma pp period 305 hyphenmath 306 $ period\acute{e} $ maticiens et physiciens de Serbie ( Vesnik ) , I − 1 ,1949 ,pp. 7 − 1 9 . ( Serbian ; Rus − open square bracket 86 closing square bracket .. Appendix to the paper of J period Hlit caron-c ijev : .. On longitudinal ribs\ centerline in a compressed{ sian place and period French summaries ) } Glas Srpske Akademije Nauka comma CXCV comma Odeljenje tehni caron-c kih nauka comma 1 comma 1 949 comma pp period[ 84 1] 7\ hyphenquad Mihailo \quad Petrovi $ \acute{c} { , } 24 . 4 . 1 868 − 8 . 636 . period 1 open 943 parenthesis . ($in Serbian closing\quad parenthesismemoriam ) \quad Glasnik \quad matemati $ \check{c} $ ko open− square bracket 87 closing square bracket .. A Survey of Elementary Mathematics period Pa r-t I : Arithmetic and Algebraf i z i period $ \check Nau c-caron{c} $ na ki kn i hyphen astronomski , ser . II , t . 3 , vol . 3 , 1 948 , pp . 123 − 1 2 7 . ( Serbian ) j iga comma Belgrade comma 1 949 comma p period 63 period open parenthesis Serbian closing parenthesis $open [ square 85 bracket ] \ddot 88 closing{U} $ square ber bracket die Beziehung .. Theory and zwischen Practice of dem the Bernsteinschen Stieltjes Integral period und Srpska Ces Akademij $ \grave a{a} $ Naukaroschen comma Limitierungsver Posebna − fahrenizdanj a . CLIV\quad commaMathematische Matemati caron-c Zeitschrift ki institut 1 , comma B . 52 Belgrade , H . comma 3 , 1 1 949 comma , pp .p period 305 − 328306 period . open parenthesis Serbian closing parenthesis [ 86open ] square\quad bracketAppendix 89 closing to the square paper bracket of .. J Complex . Hlit Numbers $ \check with{c Applications} $ i j e v to : Elementary\quad On Geometry longitudinal period ribs in a compressed place . NauGlas caron-c Srpske na knj Akademije iga comma BelNauka hyphen , CXCV , Odeljenje tehni $ \check{c} $ kih nauka , 1 , 1 949 , pp . 17 − 36grade . ( comma Serbian 1 950 ) comma p period 1 60 period open parenthesis Serbian closing parenthesis [ 87 ] \quad A Survey of Elementary Mathematics . Pa $ r−t $ I : Arithmetic and Algebra . Nau $ \check{c} $ na kn − j iga , Belgrade , 1 949 , p . 63 . ( Serbian )

[ 88 ] \quad Theory and Practice of the Stieltjes Integral . Srpska Akademij a Nauka , Posebna izdanj a CLIV , Matemati $ \check{c} $ ki institut 1 , Belgrade , 1 949 , p . 328 . ( Serbian )

[ 89 ] \quad Complex Numbers with Applications to Elementary Geometry . Nau $ \check{c} $ na knj iga , Bel − grade , 1 950 , p . 1 60 . ( Serbian ) Jovan Karamata .... 2 7 \noindentopen squareJovan bracket Karamata 90 closing\ squareh f i l lbracket2 7 .. Eine elementare Herleitung des Desarguesschen Satzes aus dem Satze von Pappos hyphen [ 90Pascal ] \ periodquad ElementeEine elementare der Mathematik Herleitung comma 5 comma des Desarguesschen 1 950 comma pp period Satzes 9 hyphen aus dem 1 0 period Satze von Pappos − Pascalopen square . Elemente bracket 9 1 der closing Mathematik square bracket , 5 .. Komplexe , 1 950 .. , zahlen pp . .. 9 in− ..1 der 0 .. . elementar .. geometrie period .. Bulletin .. de .. la .. Societ acute-e .. des [ 9math 1 ] acute-e\quad maticiensKomplexe et des\quad physicienszahlen de Macedonie\quad in open\quad parenthesisder \quad Skopjeelementar closing parenthesis\quad commageometrie t period . ..\quad 1 B u l l e t i n \quad de \quad l a \quad S o c i e t comma$ \acute .. 1{ 950eJovan} comma$ Karamata\quad pp perioddes 55 hyphen 81 period 2 7 mathopen parenthesis $ \acute[ 90{ Serbian ] e} Eine$ semicolon elementarematiciens German Herleitung et dessummary des physiciens Desarguesschen closing parenthesis de Satzes Macedonie aus dem Satze ( Skopje von Pappos ) - , Pascal t . \quad 1 , \quad 1 950 , pp . 55 − 81 . (open Serbian square. ;Elemente bracket German 92 der summary closing Mathematik square ) , 5 bracket , 1 950 .., pp Compl . 9 - 1 acute-e 0 . ment a-grave un th acute-e or e-grave me de Hadwiger period .. Glas Srpske[ 9 1 Akademije ] Komplexe Nauka comma zahlen CXCVIII in dercomma elementar geometrie . Bulletin de [ 92n period ] \quad sla periodCompl Societ 3 commae´ $ des\acute 1 950 math comma{ee´}maticiens$ pp ment period et des $ 149 physiciens\grave hyphen{ dea 1} Macedonie 61$ period un th ( open Skopje $ parenthesis\ )acute , t .{ 1e Serbian} ,$ 1 950 or semicolon $ \grave French{e} $ mesummary de Hadwiger closing, pp parenthesis . 55 . -\ 81quad . ( period SerbianGlas Commentarii ; Srpske German summary Akademije mathematici ) Nauka , CXCVIII , n.helvetici s . comma3 ,1950[ 92 25 ] commaCompl ,pp. 1 95e´ ment 149 1 comma−a` 1un pp 61 th period .e´ or ( 64Serbiane` hyphenme de Hadwiger 70; French period . open summaryGlas parenthesis Srpske ) Akademije . French Commentarii closingNauka , parenthesis mathematici helveticiopen squareCXCVIII , bracket 25 , , 93 1n . closing 95 s . 3 1 , 1 square, 950 pp , pp .bracket . 64 149− - .. 170 Sur61 . . ce( (Serbian r-t French ains ; Frenchcas ) pa summary to the power ) . Commentarii of t-r iculiers mathematici du premier th acute-e or e-grave me dehelvetici la moyenne , 25 , period 1 95 1 Bulletin, pp . 64 de- 70 la . ( French ) t−r [ 93Soci ] acute-e\quad t[Sur 93 e-acute ] ceSur des $ce math r−r −tt $acute-eains ains cas maticienspa casiculiers $paˆ et physiciens du{ premiert−r de th}$ Serbiee´ or iculiers opene` me deparenthesis la du moyenne premier Vesnik . Bulletin th closing $ \ parenthesisacute{e} $ commaor $ I\ commagravede la{ 3e hyphen} Soci$e´ met 4 commae´ dedes lamath 1 moyenne 949e´ maticiens comma . pp etBulletin period physiciens 83 hyphende de Serbie la ( Vesnik ) , I , 3 - 4 , 1 949 , pp . 83 - Soci1 3 period $ \1acute open 3 . ( parenthesis Serbian{e} $ ; French t Serbian $ summary\acute semicolon ){e} French$ des summary math closing $ \acute parenthesis{e} $ maticiens et physiciens de Serbie ( Vesnik ) , I , 3 − 4 , 1949 ,pp . 83 − t−r 1open 3 . square ( Serbian bracket[ 94 ] ; 94Le French closing d e´ veloppement square summary bracket et ) l ’ ..impo Le d acute-eance veloppementde la th e´ orie et des l quoteright s e´ ries divergentes impo to the dans power of t-r ance er e` de la th acute-el ’ orieanalyse des s math e-acutee´ riesmatique divergentes . Comptes dans rendus l quoteright du 1 congr analyses des math e´ maticiens et physiciens [ 94math ] acute-e\quadde YougoslaviematiqueLe d $period ,\ Bledacute Comptes , 1{ 949e} ,$ renduspp . veloppement 3 - du 23 .1 ( to Serbian the power et ; French l of ’ e summary r$ congr impo )to ˆ the{ t power−r }$ of e-grave ance de s des la math th acute-e $ \acute{e} $ maticienso r i e des et s physiciens $[ 95\acute ] de Sur{e} la formule$ ries des divergentes accroissements finis dans . Zbornik l ’ analyse radova Srpske akademije nauka , 7 , mathYougoslavie $ \Matematiacute comma{e} Bledcˇ ki$ institut comma matique , 1 (949 Recueil . comma Comptes des pp Travaux period rendus ) , 3 1 hyphen 9du 5 1 , $ pp 23 1 . period 1ˆ{ 1 9e - open1 24 r . parenthesis} ( Serbiancongr ; French Serbian ˆ{\grave semicolon{e}} French$ s des math summary$ \acute closing{e} $ parenthesis maticiens et physiciens de Yougoslavieopen square bracket , Bled 95 ,closing 1 949 square , pp bracket . 3 − ..23 Sursummary) . la ( formule Serbian des accroissements ; French summary finis period ) Zbornik radova Srpske akademije nauka[ comma 96 ] 7Sur comma une interpr e´ tation de la convergence de M . Milankovi c´ relative aux s e´ ries ge´ [ 95Matemati ] \quad caron-cSur ki la institut formule comma des 1 open accroissements parenthesis Recueil finis des Travaux . Zbornik closing radova parenthesis Srpske comma akademije 1 9 5 1 comma nauka pp , 7 , Matematiom $ \e´check− triques{c} . $Zbornik ki radovainstitut Srpske , akademije 1 ( Recueil nauka , 7 des , Matemati Travauxcˇ ki ) institut , 1 ,9 1 5 ( Recueil 1 , pp . 1 1 9 − 1 24 . ( Serbian ; French period 1 1 9 hyphen 1 24 perioddes Travaux open parenthesis ) , 1 95 1 , ppSerbian . 1 25 semicolon - 1 34 . ( Serbian French ; French summary ) summary closing parenthesis¨ \ [ summary[97] ) \ ] U ber die asymptotische Formel f u¨ r die Legendresche Polynome , ( mit M . Tomi c´). open square bracket 96Zbornik closing squareradova Srpske bracket akademije .. Sur une nauka interpr , 7 , acute-eMatemati tationcˇ ki institutde la convergence , 1 ( Recueil de M des period Milankovi c-acute relative aux s acute-e riesTravaux g to the) , 1 power 95 1 , ofpp e-acute . 64 - 72 om . ( acute-e Serbian hyphen; German summary ) triques period Zbornik radova Srpske akademije nauka comma 7 comma Matemati caron-c ki institut comma 1 open [ 96 ] \quad [Sur 98 ] uneProblems interpr and exercises $ \acute . Bulletin{e} $ de la tation Soci e´ t dee´ des la math convergencee´ maticiens et de physiciens M . Milankovi de $ \acute{c} $ parenthesis RecueilSerbie ( Vesnik ) , I , 1 949 . ( Serbian ) relativedes Travaux aux closing s $ parenthesis\acute{e comma} $ 1 r i95 e s 1 comma $ g ˆpp{\ periodacute 1 25{e hyphen}}$ 1 om 34 period $ \acute open parenthesis{e} − $ Serbian semicolon triques . Zbornik[ 99 ] Sur radova une notion Srpske de continuit akademijee´ r e´ naukaguli e` ,re 7 avec , application Matemati aux $ s \e´checkries de{ Fourierc} $ . ki institut , 1 ( Recueil French summaryPro closing - ceedings parenthesis of the International Congress of Mathematicians , Cambridge , USA , 1 950 , I , pp . 4 open square bracket 97 closing square bracket U-dieresis ber die asymptotische Formel f dieresis-u r die Legendresche \ centerline1{ 6des . Travaux ) , 1 95 1 , pp . 1 25 − 1 34 . ( Serbian ; French summary ) } Polynome comma[ 1 0 ..] openSur parenthesis ce r − t ains mit d Me´ periodveloppements Tomi acute-c asymptotiques closing avec parenthesis application period aux polynomes de Leg - Zbornik radovaendre Srpske . Publications akademije de naukal ’ Institut comma math 7 commae´ matique Matemati de l ’ Acad caron-ce´ mie ki serbe institut , 4 , comma 1 952 , pp 1 .. . open 69 - 88 parenthesis . Recueil des\ hspace ∗{\ f i l l } $ [ 97 ] \ddot{U} $ ber die asymptotische Formel f $ \ddot{u} $ r die Legendresche Polynome , \quad ( mit M . Tomi $ \acute{c[} 1 1 ] )Ein . Satz $ u¨ ber die Abschnitte einer Potenzreihe . Mathematische Zeitschrift , B . 56 , H . Travaux closing2 , 1 95 parenthesis 2 , pp . 2 1 comma9 - 222 . 1 95 1 comma pp period 64 hyphen 72 period open parenthesis Serbian semicolon German summary closing parenthesis \ hspace ∗{\[f 1 i l 2 l } ] ZbornikSur radova les m Srpskee´ thodes akademije d ’ extrapolation nauka , 7 , des Matemati suites $ \check temporelles{c} $ ki institut , 1 \quad ( Recueil des open squarestationnaires bracket 98 . closingRapport squarepr brackete´ sent e´ ..au Problems CCIR , Gen ande` exercisesve , 1 952 period , p . 1 Bulletin 3 . de la Soci acute-e t e-acute des math acute-e maticiens et physiciens de \ centerline[ 1{Travaux 3 ] Sur la ) sommation , 1 95 1des , s ppe´ ries . 64 de Fourier− 72 ,. (( Serbian avec M . Tomi ; Germanc´). Glas summary Srpske Akademije ) } Serbie open parenthesisnauka Vesnik , CCVI closing , n . sparenthesis . 5 , 1 953 , comma pp . 89 I - comma1 26 . ( Serbian1 949 period ; French open summary parenthesis ) Serbian closing paren- thesis ¨ t−r [ 98 ] \quad[14] ProblemsU ber das asymptotische and exercises Verhalten . Bulletin der Folgen de die durch la Soci Iteration $ definie\acute{e}sind$ . tZbornik $ \acute{e} $ open squareradova bracket Srpske 99 closing akademije square nauka bracket , 25 , ..Matemati Sur une notioncˇ ki institut de continuit , 3 ( Recueil acute-e des r e-acute guli grave-e re avec application auxdes s math e-acute $ries\acute de Fourier{e} period$ maticiens Pro hyphen et physiciens de Serbie ( Vesnik ) , ITravaux , 1 949 ) , 1 . 953 ( , Serbian pp . 45 - 60 ) . ( Serbian ; German summary ) ceedings of[ 1the 5 ] InternationalRemarque relative Congressa` ofla Mathematicianssommation des s commae´ ries de Cambridge Fourier par comma le proc USAe´ d commae´ de N 1o¨ 950rlund comma I comma pp. periodProceeding 4 1 6of period the International Congress of Mathematicians , Amsterdam , Vol . II . ( abstract ) [ 99open ] square\quad bracketSur une 1 0 closing notion square de continuit bracket .. Sur $ ce\ r-tacute ains{ de e-acute} $ r veloppements $ \acute{ asymptotiquese} $ g u l i avec $ application\grave{e} $ re avec applicationpp . 1 26 - 1 27 aux . Publications s $ \acute Scientifique{e} de$ l ’ ries Universit dee´ Fourierd ’ Alger , . s Proe´ rie− A : Sciences math . vol aux polynomes. I de , fasc Leg . hyphen I , 1 954 , pp . 6 - 1 3 . ceedingsendre period of Publications the International de l quoteright Congress Institut math of Mathematicians acute-e matique de l , quoteright Cambridge Acad , e-acute USA , mie 1 950serbe , comma 4 commaI , pp 1 952 . 4comma 1 6 pp . period 69 hyphen 88 period \noindentopen square[ bracket 1 0 ] 1\quad 1 closingSur square ce bracket $ r−t .. $ Ein a Satzi n s u-dieresisd $ \acute ber die{e Abschnitte} $ veloppements einer Potenzreihe asymptotiques period Mathe- avec application aux polynomes de Leg − matischeendre Zeitschrift . Publications comma B de period l ’ 56 Institut comma math $ \acute{e} $ matique de l ’ Acad $ \acute{e} $ mieH serbeperiod 2 , comma 4 , 1 1 95952 2 comma , pp .pp period 2 1 9 hyphen 222 period 69open− 88 square . bracket 1 2 closing square bracket .. Sur .. les .. m acute-e thodes .. d quoteright extrapolation .. des .. suites .. temporelles .. stationnaires period .. Rapport \noindentpr to the power[ 1 1 of ]acute-e\quad sentEin e-acute Satz au CCIR $ \ddot comma{u} Gen$ grave-e ber die ve comma Abschnitte 1 952 comma einer p period Potenzreihe 1 3 period . Mathematische Zeitschrift , B . 56 , H.2,1952,pp.219open square bracket 1 3 closing square− 222 bracket . .... Sur la sommation des s acute-e ries de Fourier comma .... open parenthesis avec M period Tomi c-acute closing parenthesis period Glas Srpske Akademije \noindentnauka comma[ 1 CCVI 2 ] comma\quad nSur period\quad s periodl e s 5\ commaquad m 1 953 $ comma\acute pp{e period} $ 89 thodes hyphen\quad 1 26 periodd ’ open extrapolation parenthesis \quad des \quad s u i t e s \quad temporelles \quad stationnaires . \quad Rapport Serbian$ pr semicolon ˆ{\acute French{e}} summary$ sent closing $ parenthesis\acute{e} $ au CCIR , Gen $ \grave{e} $ ve,1952,p.13. open square bracket 1 4 closing square bracket U-dieresis ber das asymptotische .. Verhalten der Folgen die durch Iteration definie\noindent to the power[ 1 3 of ] t-r\ h sind f i l period l Sur la sommation des s $ \acute{e} $ ries de Fourier , \ h f i l l ( avec M . Tomi $ \Zbornikacute{ radovac} Srpske) . akademije $ Glas nauka Srpske comma Akademije 25 comma Matemati caron-c ki institut comma 3 open parenthesis Recueil des \ centerlineTravaux closing{nauka parenthesis ,CCVI comma , n . 1 s953 . comma 5 , 1 pp 953 period , pp 45 . hyphen 89 − 601 26 period . ( open Serbian parenthesis ; French Serbian summary semicolon ) } German summary closing parenthesis \noindentopen square$ bracket [ 1 1 5 closing 4 ] square\ddot bracket{U} ..$ Remarque ber das relative asymptotische grave-a la sommation\quad desVerhalten s e-acute ries der de FourierFolgen par die durch Iteration le$ proc de f to i the n i e power ˆ{ t of−r acute-e}$ d sind e-acute . de N dieresis-o rlund period ZbornikProceeding radova of the International Srpske akademije Congress of nauka Mathematicians , 25 , Matemati comma Amsterdam $ \check comma{c} Vol$ period ki institut II period , 3 ( Recueil des open parenthesis abstract closing parenthesis pp period 1 26 hyphen 1 27 period Publications Scientifique de l quoteright Universit\ centerline acute-e{Travaux d quoteright ) ,Alger 1 953 comma , pp s e-acute . 45 rie− A60 : . ( Serbian ; German summary ) } Sciences math period vol period I comma fasc period I comma 1 954 comma pp period 6 hyphen 1 3 period \noindent [ 1 5 ] \quad Remarque relative $ \grave{a} $ la sommation des s $ \acute{e} $ ries de Fourier par le $ proc ˆ{\acute{e}}$ d $ \acute{e} $ de N $ \ddot{o} $ rlund . Proceeding of the International Congress of Mathematicians , Amsterdam , Vol . II . ( abstract ) pp . 1 26 − 1 27 . Publications Scientifique de l ’ Universit $ \acute{e} $ d ’ Alger , s $ \acute{e} $ r i e A : Sciences math . vol . I , fasc . I , 1 954 , pp . 6 − 1 3 . 28 .... M period Tomi acute-c \noindentopen square28 bracket\ h f i l 1 l 6M closing . Tomi square $ bracket\acute ..{c Sur} $ un lemme de Me r-t sub ens relatif aux nombres premiers period Comptes rendus du 79 to the power of e \noindentCongr grave-e[ 1 s 6 des ] Soci\quad e-acuteSur t acute-e un lemme s savantes de Me comma $ r Alger−t { commaens 1}$ 954 relatifcomma pp aux period nombres 277 hyphen premiers 284 period . Comptes rendus du $ 79open ˆ{ squaree }$ bracket 1 7 closing square bracket .. Evaluation acute-e l e-acute mentaire des sommes typiques de Riesz de ceCongr r-t aines $ fonctions\grave arit{e} hyphen$ s des Soci $ \acute{e} $ t $ \acute{e} $ s savantes , Alger , 1 954 , pp . 277 − 284 . hm acute-e tiques period .. Publications de l quoteright Institut math e-acute matique de l quoteright Acad acute-e mie serbe\noindent comma28 7[ comma 1 7 ] 1\ 954quad commaEvaluation $ \acute{e} $ l $ \acute{e} $ mentaireM . Tomi desc´ sommes typiques de Riesz de ce $ r−t $ aines fonctions arit − e pp period[ 1 1 hyphen 6 ] Sur 40 un period lemme de Me r − tens relatif aux nombres premiers . Comptes rendus du 79 Congr hmopen $ square\acutee` s bracket des{e} Soci$ 1e´ 8 t closing i qe´ us e savantes s square . \quad , AlgerbracketPublications , 1 .. 954 Suite , pp de . 277 fonctionelles - de 284 l . ’ Institut lin acute-e aires math et facteurs $ \acute de{ convergencee} $ matique des s de l ’ Acad e-acute$ \acute ries{ dee[} 1Fourier$ 7 ] mieEvaluation period serbe ,e´ l 7e´ ,mentaire 1 954 des , sommes typiques de Riesz de ce r − t aines fonctions arit - ppJournal . 1 − dehm40 Math .e´ tiques acute-e . matiquesPublications pures et de appliqu l ’ Institut e-acute math ese´ commamatique 35 de comma l ’ Acad 1e´ 956mie comma serbe , pp 7 , period1 954 , 87 hyphen 9 5 period pp . 1 - 40 . \noindentopen square[[ 1 bracket 81 ] 8Suite ] 1\quad 9 de closing fonctionellesSuite square de lin bracket fonctionellese´ aires .... et Sur facteurs la sommation de lin convergence $ des\acute s des acute-e s {e´}ries ries$ de de Fourier aires Fourier . et desJournal facteurs fonctions continues de convergence des s comma$ \acute ....{ openede} Math$ parenthesis riese´ matiques de avec Fourier puresM period et appliqu . Tomi c-acutee´ es , 35 closing , 1 956 parenthesis , pp . 87 - 9 period 5 . JournalPublications de[ 1 Mathde 9 ] l quoterightSur $ la\acute sommation Institut{e} mathdes$ s matiques acute-ee´ ries de matique Fourier pures de des l et fonctionsquoteright appliqu continues Acad $ e-acute\ ,acute( mie avec{e} serbe M$ . Tomi comma esc´ ,35). 8 comma ,1956 1 955 ,pp . 87 − 9 5 . comma pp periodPublications 1 23 hyphen de 1 l 38 ’ Institut period math e´ matique de l ’ Acad e´ mie serbe , 8 , 1 955 , pp . 1 23 - 1 38 . \noindentopen square[[ 1 bracket 1 1 0 9 ] D ] 1\ e´ 1hmonstration 0f i closing l l Sur square d la ’ une sommation bracket propri ....e´ t D des acute-ee´ du s co $monstratione´\fficientacute collectif{e d} quoteright$ de riescorr unee´ delation propri Fourier , ( e-acute avec des t acute-e fonctions du continues , \ h f i l l ( avec M . Tomi co$ \ e-acuteacute fficient{cB} . Ivano collectif) - . de $ corr acute-e lation comma open parenthesis avec B period Ivano hyphen vi acute-c closing parenthesisvi periodc´). Statisti Statisticˇ ka c-caron revij a ka , 5 revij , no a . comma4 , 1 955 5 , comma pp . 309 no - 3 period 1 3 . 4 comma 1 955 comma pp period 309\ hspace hyphen∗{\ 3[ 1f 1 i 3 1 l 1periodl } ] PublicationsSur la majorabilit dee´ lC ’ des Institut suites de nombres math r $e´ els\acute , ({ avece} P$ . Erd matiquehungarumlaut de l− ’o Acad $ \acute{e} $ mieopen serbe squares , ) bracket8 . Publications , 1 1955 1 1 , closing l ’ pp Institut . square 1 math 23 bracket−e´ 1matique 38 .. . Sur de l la ’ Acad majorabilite´ mie serbe acute-e , 1 0 .. , 1C 956 des , suitespp . 37 de - 52 nombres . r e-acute els t−r comma .. open[ 1 parenthesis 1 2 ] Fonctions avec Pacompo` period Erdement hungarumlaut-o r e´ gulier et s l closing ’ int e´ parenthesisgral de Frullani period , Publications( avec S . Alj an cˇ \noindentl quoterighti[c´ Institut 1). 1 0 math ] \ h acute-e f i l l D matique $ \acute de l{ quoterighte} $ monstration Acad e-acute mie d ’ serbe une comma propri 1 0 comma$ \acute 1 956{e} comma$ t pp $ \acute{e} $ perioddu co 37 $ hyphen\acute 52{ periode} $Zbornik fficient radova collectifSrpske akademije de nauka corr , 5 $ 0 ,\ Matematiacute{e}cˇ ki$ institut lation , 5 ( Recueil , ( avec des B . Ivano − open square bracket 1 1 2 closingTravaux square ) , 1 956 bracket , pp . .... 239 Fonctions - 248 . ( Serbian .... grave-a ; French compo summary to the ) power of t-r ement r acute-e gulier et\ centerline l quoteright[ 1{ int 1v i3 e-acute ] $ Introduction\acute gral de{c Frullani} a` une) th comma .$e´ orie .... deStatisti openla croissance parenthesis $ des\check fonctions avec{ Sc} rperiod$e´ elles ka .... . revij Alj anMathematikai acaron-c , 5 i , c-acute no . 4closing , 1955 ,pp . 309 − 3 1 3 . } parenthesis periodLapok , 7 , 1 956 , pp . 207 - 2 1 1 . Bulletin math e´ matique de la Soci e´ t e´ des sciences ma - th e´ \noindentZbornik radovamatiques[ 1Srpske 1 1 et ]physiques akademije\quad deSur nauka Roumanie la comma majorabilit , 1 ( 5 49 0 comma ) , no . 3Matemati $ , 1\ 957acute , pp caron-c{ .e 295} $ ki- 302 institut\quad . commaC des 5 suitesopen parenthesis de nombres Recueil r des$ \acute{e[} 1$ 14 ] eSur l s , les\quad inversions( avec asymptotiques P . Erd $ des hungarumlaut produits de−o convolutions $ s ) . . PublicationsGlas Srpske lTravaux ’ Institut closingAkademije math parenthesis nauka$ \acute comma , CCXXVIII{e 1} 956$ comma matique , n pp . s period. de 1 l 3 239 ,’ Acad hyphen 1 957 , $ 248\ ppacute period .{ 23 opene} - 59$ parenthesis . mie ( Serbian serbe Serbian ) , 10 semicolon , 1 956 , pp . 37 − 52 . French summary. closing Bulletin parenthesis de l ’ Acad e´ mie serbe des sciences , S e´ rie A : Sciences math e´ matiques , t . 20 , 1 \noindentopen square957[ bracket 1 , pp 1 .2 1 1 ] 1\ -3h 32 closing f .i l ( l FrenchFonctions square summary bracket\ )h .. f Introduction i l l $ \grave grave-a{a} une thcompo e-acute ˆ{ oriet− der la}$ croissance ement des r fonctions $ \acute{e} $ rgulier acute-e elles et[ l period 1 1’ 5 int ] .. MathematikaiSur $ les\acute facteurs{e} de$ convergence gral de uniforme Frullani des s ,e´ ries\ h f de i l Fourier l ( avec . Revue S . de\ lah f Facult i l l Alje´ an $ \check{c} $ iLapok $ \acute commades{c Sciences} 7 comma) de 1 l . 956’ Universit $ commae´ ppd ’period Istambul 207 , hyphenS e´ rie A 2 1, 22 1 period , 1 957 Bulletin , pp . 35 math - 43 . acute-e matique de la Soci e-acute e´ t acute-e des sciences[ 1 1 6 ] maSur hyphen les proc d e´ s de sommation intervenant dans la th e´ orie des nombres . Colloque \ hspaceth acute-e∗{\sur matiquesf i l la l } thZbornike´ etorie physiques des radova suites de tenu Roumanie Srpskea` Bruxelles comma akademije 1 957 1 open , Paris naukaparenthesis et Louvain , 5 49 ,0 closing1 958 , Matemati , pp parenthesis . 1 2 - 31 $ comma . \check no{ periodc} $ 3 comma ki institut , 5 ( Recueil des e` 1 957 comma[ pp 1 1 period 7 ] 295Divergence hyphen 302 de la period s e´ rie harmonique d ’ apr s Mengoli ( 1 650 ) . L ’ Enseignement \ centerlineopen squareMath{ bracketTravauxe´− 1 14 ) closing , 1 956 square , bracket pp . 239 .... Sur− 248 .... les . .... ( inversions Serbian .... ; Frenchasymptotiques summary .... des ) ....} produits .... de .... convolutions period .... Glas .... Srpskematique , 5 ( 2 ) , 1 9 60 , pp . 86 - 88 . \noindentAkademije[[ nauka 1 1 1 8 1 ] comma 3Sur ] \quad les.. CCXXVIII syst Introductione` mes comma num e´ ..riques n period$ \ degrave Cantor s period{a .} ..Zbornik$ 1 3 une comma radova th .. Srpske 1$ 957\acute comma akademije{e ..} ppnauka$ period orie , .. de 23 hyphen la croissance des fonctions r 59$ \ periodacute ..{ opene} $ parenthesis69 e l l, e Matemati s . Serbian\quadcˇ ki closing institutMathematikai parenthesis , 8 ( Recueil period des Travaux .. Bulletin ) , 1 9de 60 , pp . 1 - 8 . ( Serbian ; French Lapokl quoteright , 7 , Acad 1 956 acute-e , pp mie . serbe 207 des− sciences2 1 1 comma . Bulletin S e-acute math rie A :$ Sciences\acute math{e} acute-e$ matique matiques de comma la Soci t period $ \acute{e} $ 20t comma $ \acute 1 957{e comma} $ ppdes period sciences ma − summary) th $ \acute{e} $ matiques et physiques de Roumanie , 1 ( 49 ) , no . 3 , 1 957 , pp . 295 − 302 . 1 1 hyphen[ 132 1 period 9 ] Compl open parenthesise´ ment aux French th e´ summaryor e` mes closing de Schur parenthesis et Toeplitz , ( avec B . Baj sˇ anski ) . open square bracket 1 1 5 closing square bracket .. Sur les facteurs de convergence uniforme des s acute-e ries de Fourier \noindent Publications[ 1 14 ] \ deh l f ’i Institutl l Sur math\ h fe i´ lmatique l l e s de\ h l f’ iAcad l l i ne´ vmie e r s serbe i o n s des\ sciencesh f i l l etasymptotiques beaux - arts , 1 4 ,\ h f i l l des \ h f i l l p r o d u i t s \ h f i l l de \ h f i l l convolutions . \ h f i l l Glas \ h f i l l Srpske period Revue1 de 960 la , Facult pp . 1 e-acute 9 - 1 1 4 . des Sciences de l quoteright Universit acute-e d quoteright Istambul comma00 S e-acuteλ rie A comma 22 comma 1 957 comma Akademije[ nauka 1 20 ] , On\quad someCCXXVIII solutions of the, \ differentialquad n . equation s . \quady (x)1 = 3f( ,x)\yquad(x), ( 1with 957 V . , Mari\quadc´). pp . \quad 23 − 59 . \quad ( Serbian ) . \quad B u l l e t i n de pp period 35 hyphenAnnual review 43 period of the Faculty of Arts and Natural Science , Novi Sad , 5 , 1 960 , pp . 41 5 - 424 . ( lopen ’ Acad square $ bracket\acute 1 1{ 6e closing} $ mie square serbe bracket des .. Sur sciences les proc to , the S power $ \ ofacute acute-e{e} d e-acute$ rie s de A sommation : Sciences intervenant math $ \acute{eSerbian} $ matiques summary ) , t . 20 , 1 957 , pp . dans la th acute-e orie des nombres period Colloque e´ 1 1 − 32[ . 1( 2 1 French ] Sur quelquessummary probl ) e` mes pos s par Ramanujan . Journal of the Indian Mathematical sur la th acute-eSociety orie , vol des . 24 suites , nos. tenu 3 - 4 a-grave , 1 960 Bruxelles, pp . 343 -1 365 957 . comma Paris et Louvain comma 1 958 comma pp period 1 2 hyphen 31 period \noindent [[ 1 1 22 1 ] 5On ] \ thequad distributionSur les of intersections facteurs of de diagonals convergence in regular polygonsuniforme . desMathemat s $ \ -acute ics {e} $ ries de Fourier . Revue de la Facult open squareResearch bracket Center 1 1 7 , closing Technical square Summary bracket Report .... , Divergence No . 281 , Madison de la s acute-e , 1 961 . rie Studies harmonique in Mathematical d quoteright apr to the power$ \acute of e-grave{e} $ s Mengoli open parenthesis 1 650 closing parenthesis period L quoteright Enseignement Math acute-e hyphen desmatique Sciences comma de 5 open l ’ parenthesis Universit 2 closing $ \acute parenthesis{e} $ comma d ’ 1 Istambul 9 60 comma , pp S period $ \acute 86 hyphen{e} 88$ period rieA, 22 ,1957 ,pp . 35 − 43 . open square bracket 1 1 8 closing square bracket .... Sur les syst grave-e mes num e-acute riques de Cantor period Zbornik radova\noindent Srpske[ akademije 1 1 6 ] nauka\quad commaSur les $ proc ˆ{\acute{e}}$ d $ \acute{e} $ s de sommation intervenant dans la th $ \69acute comma{e} Matemati$ orie caron-c des ki nombres institut comma . Colloque 8 open parenthesis Recueil des Travaux closing parenthesis comma 1 9 60 commasur l pp a thperiod $ 1\ hyphenacute{ 8e period} $ openorie parenthesis des suites Serbian tenu semicolon $ \grave French{a} $ Bruxelles 1 957 , Paris et Louvain , 1 958 , pp . 1 2 − 31 . summary closing parenthesis \noindentopen square[ bracket 1 1 7 1 ] 1\ 9h closing f i l l squareDivergence bracket .. de Compl la s acute-e $ \acute ment aux{e} th$ e-acute rie or harmonique grave-e mes de d Schur ’ $ et apr Toeplitz ˆ{\grave{e}}$ commas Mengoli open parenthesis ( 1 650 ) avec . LB period ’ Enseignement Baj s-caron anski Math closing $ \ parenthesisacute{e} period − $ Publications de l quoteright Institut math acute-e matique de l quoteright Acad e-acute mie serbe des sciences et beaux hyphen arts comma\ centerline 1 4 comma{ matique 1 960 comma , 5 ( 2 ) , 19 60 , pp . 86 − 88 . } pp period 1 9 hyphen 1 1 4 period \noindentopen square[ bracket 1 1 8 1 ] 20\ h closing f i l l squareSur les bracket syst .. On $ some\grave solutions{e} $ of the mes differential num $ equation\acute{ ye} to$ the powerriques of primede Cantor . Zbornik radova Srpske akademije nauka , prime open parenthesis x closing parenthesis = f open parenthesis x closing parenthesis y to the power of lambda open parenthesis\ hspace ∗{\ x closingf i l l }69 parenthesis , Matemati comma open$ \check parenthesis{c} $ with ki V institutperiod Mari , acute-c 8 ( Recueilclosing parenthesis des Travaux period ) , 1 9 60 , pp . 1 − 8 . ( Serbian ; French Annual review of the Faculty of Arts and Natural Science comma Novi Sad comma 5 comma 1 960 comma pp period \ [41 summary 5 hyphen 424 ) period\ ] open parenthesis Serbian summary closing parenthesis open square bracket 1 2 1 closing square bracket .. Sur quelques probl grave-e mes pos to the power of e-acute s par Ramanujan period Journal of the Indian Mathematical \noindentSociety comma[ 1 vol1 9 period ] \quad 24 commaCompl nos period $ \acute 3 hyphen{e} $ 4 comma ment 1 aux 960 comma th $ pp\acute period{e 343} $ hyphen or 365 $ \ periodgrave{e} $ mesopen de square Schur bracket et Toeplitz 1 22 closing , ( square avec bracket B . Baj .. On $ the\check distribution{ s } $ of intersections anski ) . of Publications diagonals in regular polygons periodde l .. ’ Mathemat Institut hyphen math $ \acute{e} $ matique de l ’ Acad $ \acute{e} $ mie serbe des sciences et beaux − arts , 14 , 1960 , ppics . Research 1 9 − Center1 1 4 comma . Technical Summary Report comma No period 281 comma Madison comma 1 961 period Studies in Mathematical Analysis and Related Topics : .. Essays in Honor of George P acute-o lya comma \noindentStanford comma[ 1 20 1 9 ] 62\quad commaOn pp someperiod solutions1 70 hyphen 1 of 74 the period differential equation $ y ˆ{\prime \prime } ( x ) = f ( x ) y ˆ{\lambda } ( x ) , ($ withV.Mari $ \acute{c} ) . $ Annual review of the Faculty of Arts and Natural Science , Novi Sad , 5 , 1 960 , pp . 41 5 − 424 . ( Serbian summary )

\noindent [ 1 2 1 ] \quad Sur quelques probl $ \grave{e} $ mes $ pos ˆ{\acute{e}}$ s par Ramanujan . Journal of the Indian Mathematical Society , vol . 24 , nos . 3 − 4 , 1 960 , pp . 343 − 365 .

\noindent [ 1 22 ] \quad On the distribution of intersections of diagonals in regular polygons . \quad Mathemat − ics Research Center , Technical Summary Report , No . 281 , Madison , 1 961 . Studies in Mathematical Analysis and Related Topics : \quad Essays in Honor of George P $ \acute{o} $ lya , Stanford , 1 9 62 , pp . 1 70 − 1 74 . Analysis and Related Topics : Essays in Honor of George P o´ lya , Stanford , 1 9 62 , pp . 1 70 - 1 74 . Jovan Karamata .... 29 \noindentopen squareJovan bracket Karamata 1 23 closing\ h square f i l l bracket29 .. Contribution aux fondements d quoteright une th acute-e orie g to the power of e-acute n acute-e rale de la croissance period Glas Srpske \noindentakademije[ nauka 1 23 i .. ] umetnosti\quad Contribution comma .. CCXLIX aux comma fondements .. n period d s ’ period une .. th 22 comma $ \acute .. 1{ 961e} comma$ o r i.. e pp $ period g ˆ{\acute{e}}$ ..n 245 $ hyphen\acute 262{e} period$ rale .. open de parenthesis la croissance Serbian closing . Glas parenthesis Srpske period akademijeBulletin .. de nauka .. l quoteright i \quad Acadumetnosti acute-e mie , ..\quad serbe ..CCXLIX des .. sciences , \quad .. etn .. . beaux s . hyphen\quad arts22 comma , \quad .. S1 e-acute 961 rie, \quad pp . \quad 245 − 262 . \quad ( Serbian ) . .. A : .. Sciences B umath l l e t i nacute-e\quadJovan matiques Karamatade \quad commal t ’ period Acad 31 $comma\acute 1 963{e} comma$ mie pp period\quad 29serbe hyphen\quad 30 perioddes open\quad29 parenthesiss c i e n cFrench e s \quad et \quad beaux − a r t s , \quad S e´ summary$ \acute closing{e[} 1$ 23 parenthesis ] r i eContribution\quad A: aux fondements\quad S c d i e ’ n une c e s th e´ orie g n e´ rale de la croissance . Glas Srpske mathopen square $ \akademijeacute bracket{e} nauka 1$ 24 closingi matiques umetnosti square , , bracket t CCXLIX . 31 .... , A , 1 contribution 963 n . s . , pp 22 to . , the 29 1 asymptotic− 96130 , . pp ( . analysis French 245 - in 262 summary lattice . ( hyphen ) ordered groups commaSerbian open parenthesis ) . with R period Bojani acute-c \noindentand M period[ 1Bulletin Vuilleumier 24 ] \ h de f iclosing l l l ’A Acad contributionparenthesise´ mie periodserbe to Mathematic des the sciencesasymptotic Research et Centeranalysis beaux - comma arts in , Technical lattice S e´ rie Summary− A ordered Report groups , ( with R . Bojani comma$ \acute No{ periodc:} $ Sciences math e´ matiques , t . 31 , 1 963 , pp . 29 - 30 . ( French summary ) 329 comma[ 1 1 24 962 ] commaA contribution p period 1 to plus the 30 asymptotic period analysis in lattice - ordered groups , ( with R . Bojani c´ andopen M . square Vuilleumier bracketand M 1. Vuilleumier 25 ) closing . 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Boj1 ani +c´). 1Mathe 9 - . $ matics Researchmatics CenterResearch comma Center Technical , Technical Summary Summary Report Report , comma No . 432 No , Madison period 432 , 1 963comma , p . Madison 58 . comma 1 963 comma p\noindent period [[ 1 1 27 26 ] ]On\quad a classOn of functions slowly of varying regular asymptotic functions behavior and , asymptotic( with R . Bojani relationsc´). Mathe , -\quad ( with R . Boj ani $ \58acute period{cmatical} ) Research . $ Center Mathe , Technical− Summary Report , No . 436 , Madison , 1 963 , p . 1 + 34 + 3. maticsopen square Research[ 1 bracket 28 ] CenterRegularly 1 27 closing ,varying Technical square functions bracket Summary and .. theOn principle a classReport of of functions equicontinuity , No . of 432 regular , ( , with Madison asymptotic B . Baj ,sˇ behavior 1anski 963 ) . comma , p . .. open parenthesis58 . withMathematics R period Research Bojani acute-c Center ,closing Technical parenthesis Summary period Report Mathe , No . 5 hyphen 1 7 , Madison , 1 964 . Publications matical ResearchRamanujan Center Inst comma . No . 1 Technical , 1 969 , pp Summary . 235 - 246 Report . comma No period 436 comma Madison comma 1 963 comma p\noindent period [[ 1 1 29 27 ] ] \quadOn theOn degree a class of approximation of functions of continuous of regular functions by asymptotic positive linear behavior operators , , \quad ( with R . Bojani $ \1acute plus 34{c plus} ( 3 with) period M . . Vuilleumier$ Mathe )− . Mathematics Research Center , Technical Summary Report , No . 52 1 , maticalopen square ResearchMadison bracket , 1 1 Center 964 28 closing, p . 1 , 1 square Technical. bracket Summary .. Regularly Report varying , functions No . 436 and ,the Madison principle of , equicontinuity1 963 , p . comma e´ open$ parenthesis1 +[ 134 30 with ] Contribution B + period 3 Baj aux . caron-s $ fondements anski d closing ’ une th parenthesise´ orie g periodn e´ rale de la croissance II , ( avec B . MathematicsBaj Research - Center comma Technical Summary Report comma No period 5 1 7 comma Madison comma 1 964 period\noindent [ 1 28sˇ anski ] \quad ) . GlasRegularly Srpske akademije varying nauka i functions umetnosti , CCLX and , the n . s principle. 26 , 1 965 , pp of . 47 equicontinuity - 73 . , ( with B . Baj $ \Publicationscheck{ s } Ramanujan$( anski Serbian Inst ) . . Bulletin period deNo l period ’ Acad 1e´ commamie serbe 1 969 des sciencescomma etpp beaux period - arts 235 , hyphen S e´ rie 246 A : Sciencesperiod Mathematicsopen square bracket Research 1 29math closing Centere´ matiques square , Technical bracket , t . 35 .... , 1 Summary967 On , the pp .degree 43 Report - 47 of . approximation( French , No summary . 5 1 of 7 )continuous , Madison functions , 1 964by positive . linearPublications operators[ 1 comma 31 Ramanujan ] A convergence Inst test . No for Fourier . 1 , series 1 969 , ( , with pp R . . 235 Boj ani− 246c´). Indian . Journal of Mathe - open parenthesismatics , with vol . M9 , periodno . 1 , Vuilleumier 1 9 67 , pp . closing43 - 47 . parenthesis period Mathematics Research Center comma Technical Summary\noindent ReportIstarska[ 1 comma 29 22 ] \ h f i l l On the degree of approximation of continuous functions by positive linear operators , No period1 52 1 10 Beogradcomma Madison Yugoslavia comma 1 964 comma p period 1 1 period ( withopen square M . Vuilleumier bracket 1 30 closing ) . square Mathematics bracket .... Research Contribution Center aux fondements , Technical d quoteright Summary une th Report acute-e orie, g to the powerNo . of 52 e-acute 1 , n Madison acute-e rale , de1 964la croissance , p . II 1 comma 1 . open parenthesis avec B period Baj hyphen caron-s anski closing parenthesis period Glas Srpske akademije nauka i umetnosti comma CCLX comma n period s period 26\noindent comma 1 965[ 1 comma 30 ] pp\ h period f i l l 47Contribution hyphen 73 period aux fondements d ’ une th $ \acute{e} $ o r i e $ g ˆ{\acute{e}}$ nopen $ \acute parenthesis{e} $ Serbian rale closing de la parenthesis croissance period II Bulletin , ( de avec l quoteright B . Baj Acad− acute-e mie serbe des sciences et beaux hyphen arts comma S e-acute rie A : Sciences \ hspacemath acute-e∗{\ f i l matiques l } $ \check comma{ s t} period$ anski 35 comma ) . 1 Glas 967 comma Srpske pp period akademije 43 hyphen nauka 47 period i umetnosti open parenthesis , CCLX French , n . s . 26 , 1 965 , pp . 47 − 73 . summary closing parenthesis \ hspaceopen square∗{\ f i bracket l l }( Serbian 1 31 closing ) square . Bulletin bracket de.. A l convergence ’ Acad $test\acute for Fourier{e} series$ mie comma serbe open des parenthesis sciences with et R beaux − a r t s , S period$ \acute Boj{ anie} acute-c$ rie closing A : parenthesis Sciences period Indian Journal of Mathe hyphen matics comma vol period 9 comma no period 1 comma 1 9 67 comma pp period 43 hyphen 47 period \ centerlineIstarska 22 {math $ \acute{e} $ matiques , t . 35 , 1 967 , pp . 43 − 47 . ( French summary ) } 1 1 0 Beograd \noindentYugoslavia[ 1 31 ] \quad A convergence test for Fourier series , ( with R . Boj ani $ \acute{c} ) . $ Indian Journal of Mathe − matics , vol . 9 , no . 1 , 1967 , pp . 43 − 47 .

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\noindent 1 1 0 Beograd Yugoslavia