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MATEMATIQKI VESNIK JOVAN KARAMATA Vo jislav Maric The great mathematician one of the inter nationally b est known Serbian scientists Jovan Karamata was b orn into a rich merchants fa mily Living in the city of Zemun on the bor ders of the Austrian and Turkish empires and Serbia several generations of the family develo p ed a ourishing business through almost three centuries Even in they acquired an century baro que mansion in which eight gene rations of the family have b een living They b elonged to the upp er class so that even the Austrian emp eror Joseph the nd when visiting Zemun in the time of the siege of Belgrade in the AustrianTurkish war stayed in their home This has b een witnessed by a huge Austrian coat of arms painted across the ceiling of his ro om bythewell known artist Dimitrije Bacevic Here the story ab out many known even famous Europ ean families rep eats the ancestors were soldiers merchants landowners etc usually scientists authors artists etc J Karamata was b orn in Zagreb on February but sp entthe ma jor part of his life until in Zemun There he completed the rst three classes of high scho ol and then was send by his father b ecause of the First World War to Lausanne Up on graduating in he returned to Belgrade enrolled the Faculty of Engineering of Belgrade University but in following his interest and talent he moved to the Faculty of PhilosophyDepartment of Mathematics where he graduated in At the b eginning he was inuenced by his professor Miha jlo Petrovic the famous Serbian mathematician who himself was a studentof celebrated French mathematicians Ch Hermite E Picard H Poincare and trans ferred from Paris to his students in Belgrade imp ortant new mathematical ideas and theories Moreover with his colleagues M Milankovic and B Gavrilovic he created an inspiring atmosphere fruitful for creativework But Karamata considered himself as a selftaught man acquiring his education by pursuing some b o oks and research pap ers by masters already at his early studentdays An outstanding p osition in this resp ect o ccupies the famous b o ok byGPolya and G Szego Aufgab en und V Maric Lehrsatze aus der Analysis As a result three months after the graduation he su bmitted his do ctoral thesis O jedno j vrsti granica slicnih o dredjenim integralima and obtained his PhD degree already by March Then he had an usual but fast academic career disturb ed only bythewar In a Ro ckefeller Foundation fellowinParis in teaching assistant at the Faculty of PhilosophyUniversity of Belgrade in assistant professor in asso ciate one and in the full professor In he b ecame a full professor at the University of Geneva Karamata however b eing by family p osition deeply ro oted to his country left it reluctantly nancial motives if any were at the tail of his decision By the communist regime he was considered as a reactionary b ourgeois capitalist not friendly to the so cialism although pretending otherwise That minimized his professional inuence at the Department and made his life frustrating so he decided to leave But he never broke ties with his country He frequently visited the Academy lectured at the Mathematical Institute the UniversityofNovi Sad and the Faculty of Economics in Belgrade and co op erated with his former students In addition claim to his early international reputation he was elected in Corresp onding Member of the Yugoslav AcademyofSciences and Arts Zagreb and in a Corresp onding Member of Regia So cietas Scientiarum Bohemica Prague In he b ecame a Corresp onding Memb er and in a Full Member of the Serbian Academy of Sciences and Arts He was one of the founders of the Mathematical Institute of Serbian Academy of Sciences and Arts in and the rst EditorinChief of b oth of the journals Matematicki Vesnik and Publications de lInstitut Mathematique In he b ecame a memb er of the Editorial Board and Director of the journal LEnseignementMathematique from Geneva His inuence to the form and content of this journal was signicant In he married Emilija Nikola jevic who gave birth to their twosonsand a daughter His wife died in After a long illness Jovan Karamata died on August in Geneva His ashes rest in Zemun The work of Jovan Karamata b elongs to the classical analysis whichheknew to p erfection in the time of its prime He had the characteristics and the p ower of the great analysts of that ep o ch By the words of M Tomic he not only shared the opinion of GH Hardy that the mathematics was young p eoples game but demonstrated it Indeed the three most famous of Karamatas pap ers b elong to the time of his youth and attract more attention to day They app eared in These three results entered several standard monographs on the sub ject even some textb o oks and we chose to present these briey among many others app earing in his scientic pap ers see Bibliography in Jovan Karamata New metho d in Taub erian theory We recall classical N Ab els direct theorem P P Put s s then s s n implies a x s x the n n latter series being convergent for jxj Ie the convergence of the series implies Ab el summability The inverse sta P as x ie x tement is in general false For example x P the series is Ab el summable but it obviously diverges But the inverse statement holds with the additional condition na n as proved n byATaub er in Feeling something imp ortant Hardy and Littlewo o d named such inverse results Taub erian theorems Taub ers pro of was quite elementary but in JE Littlewood proved the following theorem whichisnow classical P Let the series a beAbel summable to a sum s and jna jM thenit n converges to s The generalization of convergence condition to to a nonsp ecialists do es not seem to b e signicant But the imp ortance of Taub erian theorems emanated from this theorem of Littlewo o d which b ecame the starting p oint of a new branchof mathematical analysisthe Taub erian theory His pro of however was extremely involved and long In spite of the attempts of a number of reputed analysts no signicant simplications were made Then Karamata published his sp ectacular pro of in two pages Ub er die HardyLittlewoodschen Umkehrung des Ab elschen Stetigkeitssatzes Math Zeit It runs as follows P Karamata rst noticed that Ab el summability of the series a to the sum s implies that for anypower g x x Z X x s g x x s g t dt x This is then also true for every p olynomial P t Then by applying Weierstrass approximation theorem Karamata concludes that holds for any bounded R n integrable function g x By cho osing x e g x x for e x n P s n s n of and zero otherwise the C summability ie that the series follows Hence by a simple Hardys lemma its convergence due to Much later in N Wielandt shared that the C summability step that is the use of Hardys lemma can b e avoided On the other hand in V Maricand M Tomic proved that lemma in a couple of lines The pro of was immediately internationally recognized R Schmidt call it an imp ortantdiscovery K Knopp said that it was a surprising pro of N Wiener V Maric named it elegant EC Titchmarsh extremely elegant etc In on the o ccasion of the th anniversary of Mathematische Zeitschrift Karamatas pap er was chosen among the rst most imp ortant ones selected among all of these published in this journal in that p erio d Nowadays the pro of can be found in the wellknown treatises of Knopp G Do etsch Titchmarsh Hardy DV Widder and J Favard The following true story is nice When visiting the St Andrews universityin Scotland I was intro duced to ET Copson the author of widely used textb o ok on the theory of complex functions He told me I haveknown only one Yugoslavian mathematicianKaramata When co op erating with Hardy at Oxford I found him one day nervously pacing up and down his oce Not resp onding to greetings he abruptly said I just received a letter from a young man in Belgrade who claims he can prove HardyLittlewo o d theorem in just two pages This is simply imp ossible To understand prop erly why such an excitement ab out a pro of one has to have in mind that at that time Taub erian theory was one of the highlights of the analysis Many famous analysts worked on that sub ject Among others Hardy Littlewo o d in England E Landau F Hausdor Knopp in Germany M Riesz L Fejer in Hungary etc The signicance and the depth of the theory is best seen from the fact that S Ikeharas Taub erian theorem contains the celebrated prime numb ers one x x ln xasx where xisthenumber of primes not exceeding x The theory culminated in by Wieners general Taub erian theorem one of the keystones of Mathematical Analysis to b e quoted b elow It contains Littlewo o ds theorem as a sp ecial case Yet Karamatas pro of has kept its imp ortance for at least two reasons Firstlytoderive Littlewo o ds theorem from the Wieners one is by no means easy Direct pro of is much more convenient Secondly and more imp ortant the pro of oers a new metho d which is then used eg for estimating the remainder R s sasitwas done byseveral authors eg AG Postnikov G Frend J n n Korevaar The theory of slowly varying functions This app ears to b e the most imp ortant achievementof Karamata The idea originated in many attempts of relaxing the convergence conditions such as and The most imp ortantofwhich in Wieners opinion were the ones byR Schmidt He was the rst to consider a group of terms of a sequence not only a single one as in and Thus he intro duced slowly oscillating sequences s n under the name sehr langsam oscillierende Folgen

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