Selected Title s i n Thi s Serie s

150 B . Ya. Levin , Lecture s o n entire functions , 199 6 149 Takash i Sakai, Riemannia n geometry , 199 6 148 Vladimi r I . Piterbarg , Asymptoti c method s i n th e theor y o f Gaussia n processe s an d fields, 199 6 147 S . G . Gindiki n and L . R. Volevich , Mixe d proble m fo r partia l differentia l equation s wit h quasihomogeneous principa l part , 199 6 146 L . Ya. Adrianova , Introductio n t o linea r system s o f differential equations , 199 5 145 A . N. Andriano v an d V. G . Zhuravlev , Modula r form s an d Heck e operators, 199 5 144 0 . V . Troshkin, Nontraditiona l method s i n mathematica l hydrodynamics , 199 5 143 V . A . Malyshev an d E. A. Minlos, Linea r infinite-particl e operators , 199 5 142 N . V. Krylov, Introductio n to the theory o f diffusio n processes , 199 5 141 A . A. Davydov, Qualitativ e theory o f control systems, 199 4 140 Aizi k I. Volpert, Vitaly A . Volpert , an d Vladimir A. Volpert, Travelin g wav e solution s o f parabolic systems , 199 4 139 I . V. Skrypnik , Method s fo r analysi s o f nonlinear ellipti c boundary valu e problems , 199 4 138 Yu . P . Razmyslov, Identitie s o f algebras an d thei r representations , 199 4 137 F . I. Karpelevich an d A. Ya . Kreinin , Heav y traffi c limit s fo r multiphas e queues , 199 4 136 Masayosh i Miyanishi , Algebrai c geometry, 199 4 135 Masar u Takeuchi, Moder n spherica l functions , 199 4 134 V . V. Prasolov , Problem s an d theorem s i n linea r algebra , 199 4 133 P . I. Naumkin and I. A. Shishmarev , Nonlinea r nonloca l equation s i n the theory o f waves, 1994 132 Hajim e Urakawa , Calculu s o f variations an d harmoni c maps , 199 3 131 V . V . Sharko , Function s o n manifolds : Algebrai c an d topologica l aspects , 199 3 130 V . V . Vershinin, Cobordism s an d spectra l sequences , 199 3 129 Mitsu o Morimoto , A n introductio n t o Sato's hype r functions, 199 3 128 V . P. Orevkov , Complexit y o f proof s an d thei r transformation s i n axiomati c theories , 1993 127 F . L. Zak, Tangent s an d secant s o f algebrai c varieties, 199 3 126 M . L . Agranovskii, Invarian t functio n space s o n homogeneou s manifold s o f Li e group s and applications , 199 3 125 Masayosh i Nagata, Theor y o f commutative fields , 199 3 124 Masahis a Adachi , Embedding s an d immersions , 199 3 123 M . A . Akivis an d B. A. Rosenfeld, Eli e Cartan (1869-1951) , 199 3 122 Zhan g Guan-Hou, Theor y o f entire and meromorphi c functions : Deficien t an d asymptoti c values and singula r directions , 199 3 121 I . B . Fesenk o an d S . V . Vostokov , Loca l fields an d thei r extensions : A constructiv e approach, 199 3 120 lakeyuk i Hid a and Masuyuki Hitsuda, Gaussia n processes , 199 3 119 M . V. Karasev and V. P. Maslov, Nonlinea r Poisso n brackets. Geometry and quantization , 1993 118 Kenkich i Iwasawa, Algebrai c functions , 199 3 117 Bori s Zilber , Uncountabl y categorica l theories , 199 3 116 G . M. Fel'dman, Arithmeti c o f probability distributions , an d characterizatio n problem s on abelia n groups , 199 3 115 Nikola i V. Ivanov , Subgroup s o f Teichmiille r modula r groups , 199 2 114 Seiz o ltd, Diffusio n equations , 199 2 (See the AM S catalo g fo r earlie r titles ) This page intentionally left blank Lectures o n Entire Function s Boris Yakovlevich LEVIN 1906-1993 10.1090/mmono/150 Translations o f MATHEMATICAL MONOGRAPHS

Volume 15 0

Lectures o n Entire Function s

B. Ya. Levin

In collaboration with

Yu. Lyubarski i M. Sodin V. Tkachenko

Qff l^^,lY & American Mathematical Societ y '5 Providence , Rhode Island B. fl. JIEBH H IIEJIHE ^YHKIIH H

ITpn ynacTH H K) . M . JIio6apcKoro , M . JI . Co^HHa , B . A . TKa^eiiK O

Translated b y Vadi m Tkachenk o fro m a n origina l Russia n manuscrip t

EDITORIAL COMMITTE E AMS Subcommitte e Robert D . MacPherso n Grigorii A . Marguli s James D . Stashef f (Chair ) ASL Subcommitte e Steffe n Lemp p (Chair ) IMS Subcommitte e Mar k I . Freidlin (Chair )

2000 Mathematics Subject Classification. Primar y 30Dxx , 30D20 .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/mmono-150

Library o f Congres s Cataloging-in-Publicatio n Dat a Levin, B . IA. (Bori s IAkovlevich ) Lectures o n entir e function s / B . Ya. Levin ; i n collaboration wit h Yu . Lyubarskii , M . Sodin , V. Tkachenko . p. cm. — (Translation s o f mathematical monographs , ISS N 0065-928 2 ; v. 150 ) Translated fro m th e author' s Russia n manuscript . Includes bibliographica l reference s an d indexes . ISBN 0-8218-0282- 8 (alk . paper ; har d cover ) ISBN 0-8218-0897- 4 (alk . paper; sof t cover ) 1. Functions, Entire . I . Title. II . Series . QA351.E5L484 199 6 96-31 8 515'.98—

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provided th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission should b e addressed t o the Acquisitions Department, America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 199 6 b y the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 09 0 8 07 0 6 0 5 Contents

Preface x i Introduction x v Part I . Entire Function s o f Finite Orde r 1 Lecture 1 . Growt h o f Entire Function s 3 1.1. Th e growt h scal e fo r entir e functions 3 1.2. Orde r an d typ e o f entire functions 3 1.3. Th e relatio n betwee n th e growt h o f a n entir e functio n an d th e decrease o f the coefficient s o f its power serie s expansion 5 Lecture 2 . Mai n Integra l Formula s fo r Function s Analyti c i n a Dis k 9 2.1. Th e Poisso n formul a an d the Schwar z formul a 9 2.2. Th e Poisson-Jense n formul a 9 2.3. Th e Jense n formul a 1 0 2.4. Th e Nevanlinn a characteristic s 1 1 2.5. Som e corollaries o f the Jensen formul a 1 3

Lecture 3 . Som e Application s o f the Jense n Formul a 1 5 3.1. A theorem o n (J)-quasianalyticit y 1 5 3.2. Th e convergenc e exponen t an d th e uppe r densit y o f the sequenc e of zeros 1 7 3.3. Completenes s o f a system o f exponential function s 1 9 3.4. Completenes s o f a special system o f functions i n countably norme d spaces 2 0

Lecture 4. Factorizatio n o f Entire Functions o f Finite Orde r 2 5 4.1. Th e Weierstrass canonica l product 2 5 4.2. Th e Hadamard theore m 2 6 4.3. Estimate s fo r canonica l products 2 8 Lecture 5 . The Connection betwee n the Growt h o f Entire Functions and th e Distribution o f their Zero s 3 1 5.1. Function s o f noninteger orde r 3 1 5.2. Function s o f integer orde r 3 2

Lecture 6 . Theorem s o f Phragmen an d Lindelo f 3 7 6.1. Function s analyti c insid e an angl e 3 7 6.2. Entir e function s wit h value s i n Banach algebra s 4 0 viii CONTENT S

6.3. Application s o f the Phragme n an d Lindelo f theorem s t o Banac h algebras 4 3

Lecture 7 . Subharmoni c Function s 4 5 7.1. Definitio n an d basi c properties 4 5 7.2. Th e F. Ries z theorem and th e Jensen formul a 4 8 7.3. Phragmen-Lindelo f theorem s fo r subharmoni c function s 4 9 7.4. Logarithmicall y subharmonic function s 5 0 Lecture 8 . Th e Indicator Functio n 5 3 8.1. Th e definitio n an d p-trigonometri c convexit y o f the indicato r 5 3 8.2. Propertie s o f trigonometrically conve x functions 5 5 8.3. Application s o f properties o f the indicato r functio n 5 8

Lecture 9 . Th e Polya Theorem 6 3 9.1. Supportin g function s o f convex sets 6 3 9.2. Th e Bore l transform an d the Poly a theorem 6 5

Lecture 10 . Application s o f the Polya Theorem 6 9 10.1. Th e Paley-Wiene r theore m 6 9 10.2. Analyti c continuatio n o f a powe r serie s 7 0 10.3. Analyti c functional s 7 3

Lecture 11 . Lower Bound s fo r Analyti c an d Subharmoni c Function s 7 5 11.1. Th e Caratheodor y inequalit y 7 5 11.2. Th e Carta n estimat e 7 6 11.3. Lowe r bounds fo r the modulus o f an analyti c functio n i n a dis k 7 9

Lecture 12 . Entir e Function s with Zero s on a Ray 8 1 12.1. Asymptoti c behavio r o f canonical product s 8 1 12.2. Theore m o n a segment o n the boundary o f the indicator diagra m 8 3 12.3. Lowe r bound fo r the canonica l produc t wit h positive zero s havin g density 8 6 Lecture 13 . Entire Function s with Zero s on a Ray (Continuation ) 9 1 13.1. Th e Valiro n theorem 9 1 13.2. Function s o f completely regula r growt h 9 4 Part II . Entire Function s o f Exponential Typ e 9 7

Lecture 14 . Integra l Representatio n o f Functions Analyti c in the Half-plane 9 9 14.1. Th e R . Nevanlinn a formul a 9 9 14.2. Representatio n o f a functio n f(z) analyti c i n the half-plan e suc h that lo g \f(z)\ admit s a positive harmonic majorant 10 1 14.3. Applicatio n to the theory o f quasianalytic classe s 10 5

Lecture 15 . The Hayman Theore m 10 9

Lecture 16 . Function s o f Class C an d thei r Application s 11 5 16.1. Propertie s o f functions o f class C 11 5 16.2. Th e Titchmarsh convolutio n theore m an d a problem o f Gelfand 11 9 16.3. Mea n periodic function s 12 1 CONTENTS i x

Lecture 17 . Zero s o f Functions o f Class C 12 5 17.1. Th e generalize d Jense n formul a 12 5 17.2. Asymptoti c propertie s o f zeros o f functions o f class C 12 6

Lecture 18 . Completenes s an d Minimalit y o f Systems o f Exponential Func - tions in L 2(a,b) 13 1 Lecture 19 . Hard y Space s i n the Upper Half-Plan e 13 7 19.1. Definitio n an d basi c properties 13 7 19.2. Boundar y value s o f functions o f H\ 13 9 19.3. M . Riesz's theorem o n conjugate harmoni c function s an d the gen - eral form o f linear functional s i n H+ 14 2 19.4. Th e Paley-Wiener theore m fo r H% 14 6

Lecture 20 . Interpolatio n b y Entire Function s o f Exponential Typ e 14 9 20.1. Space s I£ an d B a 14 9 20.2. Interpolatio n theore m wit h intege r node s 15 0 20.3. Interpolatio n i n the space s L£, 1 < p < oo 5 with intege r node s 15 1 Lecture 21. Interpolation b y Entire Function s fro m th e Space s L^ an d B^ 15 5 21.1. Interpolatio n b y function s fro m B^ an d L n 15 5 v 21.2. Interpolatio n b y function s fro m L 0 with a

Lecture 23 . Ries z Base s Forme d b y Exponential Function s i n L 2(—7r, TT) 16 9 23.1. Definitio n an d propertie s o f Riesz bases 16 9 23.2. Th e 1/4-theore m 17 2 Appendix. Completenes s o f the Eigenfunctio n Syste m o f a Quadratic Oper - ator Penci l 18 1 Al. Twofol d completenes s o f the system /C a 18 1 A2. Completenes s o f the system K% 18 3

Part III . Som e Additional Problem s o f the Theor y o f Entire Function s 18 5

Lecture 24 . Th e Formula s o f Carleman an d R . Nevanlinn a an d thei r Appli - cations 18 7 24.1. Th e Carlema n formul a 18 7 24.2. Th e Phragmen-Lindelo f principl e a s formulated b y F. an d R . Ne - vanlinna 19 0 24.3. R . Nevanlinna's formul a fo r a half-disk 19 2

Lecture 25 . Uniquenes s Problem s fo r Fourie r Transform s an d fo r Infinitel y Differentiable Function s 19 5 25.1. Uniquenes s theorem fo r Fourie r transform s 19 5 25.2. Constructio n o f entire function s decayin g o n the rea l axi s 19 9 25.3. Uniquenes s proble m o f Gelfan d an d Shilo v fo r infinitel y differen - tiable function s 20 4 x CONTENT S

Lecture 26 . Th e Matsae v Theore m o n the Growt h o f Entire Function s Ad - mitting a Lowe r Bound 20 9 26.1. A lower bound fo r harmonic functions o f order greater than on e in the uppe r half-plan e 20 9 26.2. Refinemen t o f the upper boun d 21 2 26.3. Proo f o f Matsaev's theore m 21 3 26.4. Entir e function s admittin g a lowe r boun d fo r p < 1 21 4 Lecture 27 . Entir e Function s o f Class P 21 7 27.1. Propertie s o f functions o f class P 21 7 27.2. Meromorphi c function s wit h interlacing zero s and pole s 22 0 27.3. Theore m o f Hermite and Biehler fo r entire functions o f exponential type 22 2 Lecture 28 . S.N. Bernstein' s Inequality fo r Entire Functions o f Exponentia l Type an d it s Generalization s 22 7 28.1. P-majorant s 22 7 28.2. Operator s preservin g inequalitie s 23 0 28.3. S . N. Bernstein's inequalit y an d Banac h algebra s 23 6

Added i n Proof 23 8

Bibliography 23 9

Author Inde x 24 5

Subject Inde x 247 Preface

Boris Yakovlevich Levi n was born o n December 22 , 190 6 in Odessa , the beau - tiful por t cit y o n the Blac k Sea , on e o f the mai n trade an d cultura l center s i n th e South o f the Russian Empire. " I am a n Odessi t b y social origin and nationality 1," he use d t o joke. Hi s fathe r wa s a cler k fo r a Blac k Se a steame r company , whos e work ofte n too k hi m to variou s beac h ports o n long-ter m missions . B . Ya . (a s h e was calle d b y hi s colleague s an d friends , rathe r tha n mor e forma l Bori s Yakovle - vich) spen t hi s yout h movin g with hi s famil y fro m on e port tow n t o another . H e kept hi s devotio n t o the se a al l hi s life , wa s a n excellen t swimme r an d longe d fo r the Blac k Sea , whil e living far fro m i t i n Kharkov . Being o f nonproletarian socia l origin , B . Ya. had n o righ t t o highe r educatio n in post-revolutionar y Russi a afte r graduatin g fro m secondar y school . Fo r som e time h e worked a s a n insuranc e agen t an d newspape r dispatcher , an d a s a welde r during th e constructio n o f oi l pipe-line s i n th e Nort h Caucasus . Thi s gav e hi m the righ t t o enlis t a s a univerisit y student , an d i n 192 8 h e starte d hi s first yea r at th e Departmen t o f Physic s an d Engineerin g o f Rosto v University , Russia . H e and his friend decided , before concentrating o n physics, to widen and improve their knowledge o f mathematics . I t wa s th e choic e o f destiny : onc e enterin g mathe - matics the y neve r parte d wit h it . Bot h becam e famou s experts : Bori s Levi n i n analysis an d Nikola i Efimo v i n geometry . T o a grea t exten t the y wer e influence d by Dmitri i Mordukhai-Boltovskoi , a n interestin g an d origina l mathematicia n wit h wide interests who worke d at th e Rosto v Universit y a t tha t time . While a second year student B . Ya. obtained hi s first mathematical resul t whe n he solved a problem propose d b y Mordukhai-Boltovskoi. H e investigated th e func - tional equatio n

where R(x) i s a give n rationa l function . Thi s equatio n generalize s th e functiona l equation T(x + 1 ) = xT(x) o f th e Eule r T-funetion . B . Ya . prove d that , apar t from som e exceptions, al l solutions t o thi s equatio n ar e hypertranscendental, jus t as T(x). Al l exceptiona l case s wer e explicitl y describe d b y him . Thi s theore m generalizes the famou s Holde r theorem . In 193 2 B . Ya . graduate d fro m th e university , an d fo r th e nex t thre e year s worked o n his dissertation an d taught mathematic s a t a technical institute i n Ros- tov. Hi s close personal and scientific friendship with Naum Akhiezer (Kharkov ) an d

1 "Social origin " (o r "clas s origin" ) an d "nationality " wer e obligator y question s i n al l ap - plication form s i n th e Sovie t Union . Unlik e i n othe r countries , th e notion s "nationality " an d "citizenship" ha d differen t meaning s there .

xi xii PREFAC E

Mark Krein (Odessa ) started at that time.2 I n 193 6 B. Ya. submitted his Candidate of Scienc e dissertatio n "O n th e growt h o f an entir e functio n alon g a ray , an d th e distribution o f it s zero s wit h respec t t o thei r arguments " t o Kharko v University , but wa s awarde d wit h th e highes t degre e o f Docto r i n Mathematics , whic h wa s an extremel y rar e event. I n this dissertation B . Ya . founde d th e genera l theory o f entire functions o f completely regular growth, whose creation he shared with Albert Pfluger. In 193 5 B. Ya. moved to Odessa and began teaching mathematics at the Odess a Institute o f Marine Engineering. I n due time he got a Chair o f Mathematics o f this institute. Paralle l to hi s teaching B . Ya. spent a lo t o f time an d effor t i n advisin g his colleagues who worked o n hydrodynamical problem s o f ships an d mechanic s o f construction. I n hi s late r year s h e woul d sa y tha t teachin g an d communicatin g with engineer s i n a seriou s technica l universit y i s a n importan t experienc e fo r a mathematician. Starting fro m th e middl e 1930 s a ne w schoo l o f functiona l analysi s ha s bee n forming aroun d Mar k Krei n i n Odessa , an d B . Ya. , a s h e late r use d t o say , ex - perienced it s strenghtenin g influence . H e becam e intereste d i n almos t periodi c functions, quasianalyti c classe s and relate d problem s o f completeness an d approx - imation, algebrai c problem s o f the theory o f entire functions , an d Sturm-Liouvill e operators. Thes e remained th e main fields o f interest durin g hi s life . In Odessa , the firs t student s o f B. Ya. have started thei r ow n research. Mosh e Livshits an d Vladimi r Potapov , wh o becam e well-know n specialist s i n functiona l analysis, wer e i n equa l measur e student s o f Mar k Krei n an d Bori s Levin . To - day, th e famil y tre e o f B. Ya.' s mathematica l children , grandchildren , an d great - grandchildren contain s mor e than a hundred mathematicians . During Worl d War II , B. Ya. worked with hi s institute i n Samarkand (Uzbek - istan) . His attempts to join active military service failed, sinc e Pull Professors wer e exempt fro m th e draft . Afte r th e wa r B . Ya . returne d t o Odessa . A t tha t tim e a destructio n o f mathematic s a t Odess a Universit y began . Mar k Krei n an d hi s colleagues wer e not permitte d t o retur n t o wor k a t th e university , an d ver y soo n an anti-semitic campaign wage d against Mark Krein and B. Ya. forced the latter t o leave Odessa . O n invitation o f Naum Akhieze r i n 1949 , B. Ya. moved to Kharkov . During severa l decades afte r th e end o f World War II, some other mathematician s moved from Odess a to Kharkov: Izrai l Glazman, Mikhai l Dolberg, Moshe Livshits, Vladimir Potapov . However , B . Ya. ha s kep t clos e ties with Odessa , Mar k Krein , and the mathematicians o f Krein's circl e fo r the whol e life . Despite al l difficulties, th e period fro m lat e fortie s to lat e sixtie s was the tim e of blossoming o f the Kharko v mathematica l school . A t that tim e Naum Akhiezer , Boris Levin , Vladimi r Marchenko , Aleksand r Povzner , an d Alekse i Pogorelo v worked i n Kharkov , an d thei r impac t determine d th e imag e o f Kharko v mathe - matics fo r man y years . Prom 1949 , B. Ya . worke d a t Kharko v University . I n additio n t o undergrad - uate course s o f calculus , theor y o f function s o f a comple x variabl e an d functiona l analysis, he taught advance d course s o n entire functions , quasianalyti c classes , al - most periodi c functions, harmoni c analysi s and approximatio n theory , an d Banac h algebras. Th e lecture s wer e distinguished b y their originality , dept h an d elegance .

2Reminiscences o f Mar k Krein , writte n b y B . Ya., wer e published i n the Ukrainia n Mathe - matical Journal, 46 , no. 3 , 1994. PREFACE xm

B. Ya . use d t o includ e hi s own , yet-unpublishe d result s a s wel l a s ne w origina l proofs o f known theorems. H e attracted a very wide audience o f students o f various levels and als o research mathematicians . Thi s boo k emerge d fro m note s o f one o f such course s and i t i s a great pit y that note s o f other course s ar e not available . In 195 6 he published his monograph "Distributio n o f zeros of entire functions" , which greatl y influence d severa l generation s o f analysts . I t wa s translate d int o German an d Englis h an d revise d i n 1980 . Eve n no w the boo k i s the mai n sourc e on the subject . During the same year B. Ya. started hi s Thursday semina r at Kharko v Univer - sity. Fo r about 4 0 years it has been a school fo r Kharko v mathematician s workin g in analysi s an d ha s bee n a cente r o f activ e mathematica l research . Th e majo r part o f seminar talk s concerne d comple x analysi s an d it s applications . Neverthe - less, ther e wa s n o restrictio n o n the subject : ther e wer e talk s o n Banac h spaces , spectral theor y o f operators , differentia l an d integra l equations , an d probabilit y theory. A meeting o f the seminar usually lasted mor e than tw o hours, with a short break. I n mos t case s detaile d proof s wer e presented . It s activ e participant s in - cluded Vladimi r Azarin , Aleksand r Eremenko , Serge i Favorov, Aleksandr Fryntov , Anatolii Grishin , Vladimi r P . Gurarii , Illic h Hachatryan , Mikhai l Kadets , Victo r Katsnelson, Vladimi r Logvinenko , Yuri i Lyubich, Vladimi r Matsaev , Iossi f Ostro - vskii, Igor 5 Ovcharenko , Victo r Petrenko , Le v Ronkin , an d man y others . B . Ya . has alway s been prou d an d delighte d wit h achievement s o f the participant s o f hi s seminar. In 1969 , without interruptin g hi s teachin g a t th e university , B . Ya. organize d and heade d th e Departmen t o f the Theor y o f Function s a t th e Institut e fo r Lo w Temperature Physics and Engineering of the Academy of Sciences of Ukraine, where he gathered a group o f his former student s an d young colleagues. H e worked ther e to the last days of his life. A well-known western mathematician working in complex analysis onc e said : "I t i s a typica l Sovie t habi t t o mak e secret s fro m everything ; evidently, "Lo w Temperature Physics " i s just a code fo r functio n theory. " The nam e o f the founde r an d first directo r o f the institut e Bori s Verki n mus t be mentione d here . A specialis t i n experimenta l physics , h e hel d mathematic s i n high esteem and gav e a lot o f support to its progress. "Mathematician s ennoble the institute," Verkin used to say. Du e to his initiative, Naum Akhiezer, Izrail Glazman, Vladimir Marchenko, Anatolii Myshkis, Aleksei Pogorelov joined the institute in the early sixtie s and ver y soo n the Mathematical Divisio n o f the institute becam e on e of the leading mathematical centers in the former Sovie t Union, with the wonderfu l creative athmospere . It i s not ou r intentio n t o giv e here a detaile d descriptio n o f mathematical ac - tivities o f B. Ya. W e only mention that h e kne w ho w to find unexpectedl y simpl e ways leadin g t o a solutio n o f a proble m whic h fro m th e beginnin g seeme d t o b e extremely complicated . Afte r hi s talks and work s one would be puzzled wh y othe r mathematicians wh o attacke d th e sam e proble m di d no t hav e th e sam e insight ? The participants o f his seminar remember that sometime s afte r somebody' s "hard " talk B . Ya . propose d hi s simpl e an d elegan t solution . A t th e sam e tim e B . Ya . mastered the fine analytic techniques, which h e successfully use d i f required. The main part o f results obtained b y B. Ya. are related to the theory o f entire functions. Bein g intereste d i n th e centra l problem s o f thi s theory , h e foun d ne w and importan t connection s wit h othe r domain s o f analysis . Hi s result s helpe d t o transfer application s o f the theory o f entire functions t o functional analysi s and th e XIV PREFACE spectral theor y o f differentia l operator s t o a deepe r level . Ofte n B . Ya . expresse d the viewpoint that th e theory o f entire function s remain s o f importance du e to it s numerous applications . Boris Levi n live d a lon g life , ful l o f mathematica l ques t an d discoveries . H e experienced many difficul t periods , but despit e all strokes o f fate remaine d faithfu l to hi s highes t mora l principle s whic h h e defende d openl y an d selflessly . H e di d not have , an d di d no t tr y t o see k favour s fro m officialdom . Unti l mid-80 s h e wa s not allowe d t o trave l abroa d an d ha d ver y scarc e possibilitie s t o contac t foreig n colleagues. I n Kharkov h e lived in a small and wet ground-floo r appartment . Nev - ertheless, ver y ofte n B . Ya. woul d invit e hi s colleague s an d student s t o hi s home. Several hour s woul d be devote d to mathematics . The n suppe r tim e woul d arrive , and hi s wif e Liya , a woma n o f great charm , kindness , an d benevolence , joined th e guests. Afte r traditiona l stron g te a whic h B . Ya. alway s mad e himself , ther e wa s the time fo r discussing politics and politicians, fo r storytelling and poetry, in which B. Ya. was the expert an d connoisseur . Outstanding mathematician , brillian t lecture r an d storyteller , witt y compan - ion, B . Ya . radiate d som e kin d o f energ y tha t attracte d t o hi m eve n peopl e wh o were ver y fa x fro m mathematics . H e wa s a perso n o f th e highes t qualit y t o th e many peopl e who knew him .

For many year s B. Ya. has planned a boo k base d o n hi s lecture cours e a t th e Moscow University i n 1969 . It wa s intended fo r a reader intereste d i n application s of the theory o f entire functions . During the las t tw o years o f his life , w e worked wit h B . Ya . on preparation o f this book. Th e material o f the lectures was shaped, extended, an d augmented wit h a bibliography . Initially , B . Ya . planne d t o includ e som e importan t application s of the theor y o f entire function s t o the spectral theory o f operators, discovere d i n various direction s b y Loui s d e Brange s an d Vladimi r Matsaev , bu t thi s tas k wa s never completed . B. Ya. hoped to see his book published both in Russian and English and worked on the manuscrip t unti l the las t day s o f his life . H e died o n August 24 , 1993 . Hi s daughter Natalya and wif e Liya had preceded him, passing awa y in 198 0 and 1992 . He is survived b y his son Mikhail and tw o grandchildren .

David Drasi n carefull y rea d th e entire manuscript an d mad e man y correction s and suggestion s relate d bot h t o mathematic s an d English . Gundorp h Kristianse n called ou r attentio n to som e misprints. W e are very grateful t o them fo r that .

Yurii Lyubarskii, Mikhai l Sodin , Vadi m Tkachenk o Introduction

This monograph originate d fro m a lecture cours e which I gav e at th e Mosco w University i n 1969 . Th e lectures were printed b y the University i n an edition o f 500 copies. The titl e o f the monograp h remain s th e same , bu t i t i s not a secon d edition , since the main part o f lectures has been written anew , some new material has bee n added, an d som e ol d topic s have been extende d resultin g i n a substantial increas e of the entir e volume . The theor y o f entire function s ha s a multitude o f application s i n calculu s an d functional analysis . I mad e i t m y goa l to presen t th e mai n fact s o f the theor y o f entire function s fro m tha t poin t o f view and tried , a s much a s possible, to develo p a connectio n betwee n application s an d th e genera l theory . I hop e tha t suc h a n exposition wil l hel p i n masterin g th e method s o f th e theor y o f entir e functions . Many section s o f thi s monograp h contai n problem s wit h application s relate d t o these topics. Thei r solutio n i s not necessar y to comprehen d subsequen t parts , bu t may be o f some use. N o special knowledg e i s required t o rea d thi s book, excep t a conventional universit y cours e on the theory o f functions o f a comple x variable . Yu. Lyubarskii, M . Sodin , an d V . Tkachenk o helpe d m e very muc h i n writin g the monograph ; i t woul d no t hav e been written withou t thei r support . I. Ostrovskii read the whole manuscript an d made several remarks, which wer e taken int o accoun t i n the final text. A . Eremenko mad e severa l usefu l remark s t o Part I . I would lik e to expres s my deep gratitude to these colleagues . This monograph wa s written with a partial support b y a grant fro m the Amer - ican Mathematical Society , which i s highly appreciated .

B. Levi n This page intentionally left blank Bibliography

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53. , Value distribution and exceptional sets, Sem . Math . Sup. , vol . 79 , Presse s Univ . Montreal, Montreal , 1982 , pp. 79-147 . 54. W . K . Hayma n an d P . B . Kennedy , Subharmonic functions. I , Academi c Press , Ne w York , 1976. 55. W . K . Hayman an d B. Kjellberg, On the minimum of a subharmonic function on a connected set, Studie s i n Pure Mathematic s (Pau l Erdos , ed.), Birkhauser, Basel , 1983 , pp. 291-322 . 56. J . R . Higgins , Five short stories about the cardinal series, Bull . Amer. Math. Soc . 1 2 (1985) , 45-89. 57. K . Hoffman , Banach spaces of analytic functions, Prentic e Hall , Englewood Cliffs , NJ , 1962 . 58. L . Hormander, A uniqueness theorem of Beurling for Fourier transform pairs, Ark . Mat . 2 9 (1991), 237-240 . 59. S . V . Hrusce v [Khrushchev] , N . K . Nikol'skii , an d B . S . Pavlov , Unconditional bases of exponentials and of reproducing kernels, Comple x Analysi s an d Spectra l Theory (Leningrad , 1979/1980), Lectur e Note s i n Math. , vol . 864 , Springer-Verlag, Berlin , 1981 , pp. 214-336 . 60. Ya . I. Khurgin and V. P. Yakovlev, Compactly supported functions in physics and engineering, "Nauka", Moscow , 1971 . (Russian ) 61. V . P. Havin [Khavin ] and B. Joricke, The uncertainty principle in harmonic analysis, Springer - Verlag, Berlin , 1994 . 62. J.-P . Kahane , Sur quelques problemes d'unicite et de prolongement, relatifs aux fonctions approchables par des sommes d'exponentielles, Ann . Inst . Fourie r (Grenoble ) 5 (1953-1954) , 39-130; Sur les fonctions moyenne-periodiques bornees, Ann . Inst . Fourie r (Grenoble ) 7 (1957), 293-314 . 63. , Lectures on mean periodic functions, Tat a Institute o f Fundamental Research , Bom - bay, 1959 . 64. Y . Katznelso n an d S . Mandelbrojt , Quelques classes de fonctions entieres. Le probleme de Gelfand et Shilov, C . R . Acad . Sci . Paris Ser . I Math. 25 7 (1963) , 345-348 . 65. V . E . Katsnelson , Exponential bases in L 2, Funktsional . Anal , i Prilozhen. 5 (1971) , no . 1 , 37-47; Englis h transl . i n Functional . Anal . Appl . 5 (1971) , no. 1 , 31-38. 66. B . N. Khabibullin, Smallness of growth on the imaginary axis of entire functions of exponen- tial type with given zeros, Mat . Zametk i 43 (1988) , no. 5 , 644-650; Englis h transl . i n Math . Notes 43 (1988) . 67. , Sets of uniqueness in spaces of entire functions of one variable, Izv . Akad . Nauk . SSSR 5 5 (1991) , no. 4 , 1101-1128 ; English transl . i n Math . USSR-Izv . 3 9 (1992) . 68. , Nonconstructive proofs of the Beurling-Malliavin theorem on the radius of complete- ness, and nonuniqueness theorems for entire functions, Izv . Ross . Akad. Nau k Ser . Mat . 5 8 (1994), 125-146 ; Englis h transl . i n Russ . Acad . Sci . Izv . Math. 4 5 (1995) . 69. A . I . Kheifits , A characterization of zeros of certain special classes of entire functions of finite degree, Teor . Funktsi T Funktsional. Anal , i Prilozhen. 9 (1969) , 3-13. (Russian ) 70. B . Kjellberg , On certain integral and harmonic functions: A study in minimum modulus, Thesis, Univ . Uppsala , Uppsala , 1948 . 71. P . Koosis , Introduction to H p spaces, Cambridg e Univ . Press , Cambridge , 1980 . 72. , The logarithmic integral. I , Cambridge Univ . Press, Cambridge, 1988 ; II, Cambridg e Univ. Press , Cambridge , 1992 . 73. J . Korevaar , Zero distribution of entire functions and spanning radius for a set of complex exponentials, Aspect s o f Contemporary Comple x Analysi s (D . A. Brannan an d J . G . Clunie , eds.), Academi c Press , Ne w York , 1980 , pp. 293-312 . 74. I . F . Krasichkov-Ternovskii , An estimate for the subharmonic difference of subharmonic functions. I , Mat . Sb . 10 2 (1977) , no . 2 , 216-247 ; II , Mat . Sb . 10 3 (1977) , no . 1 , 69-111 ; English transl . i n Math. USSR-Sb . 3 1 (1977) , 191-218 ; 32 (1977) , 59-97 . 75. , An interpretation of the Beurling-Malliavin theorem on the radius of completeness, Mat. Sb . 180 (1989) , no. 3, 397-423; English transl . i n Math. USSR-Sb . 66 (1990) , 405-429. 76. M . G . Krein , Fundamental aspects of the representation theory of Hermitian operators with deficiency indices (m,n), Ukrain . Mat . Zh . 1 (1949) , no . 2 , 3-65 ; Englis h transl . i n Amer . Math. Soc . Transl. Ser . 2 97 (1971) . 77. , On the indefinite case of the Sturm-Liouville boundary problem in the interval [0 , oo), Izv. Akad . Nau k SSS R 1 6 (1952) , no . 5 , 293-324. (Russian ) 78. N . S . Landkof , Foundations of the modern potential theory, Springer-Verlag , Berlin , 1972 . 79. A . F . Leont'ev , Exponential series, "Nauka" , Moscow , 1976 . (Russian ) 242 BIBLIOGRAPHY

80. B . Ya . Levin , On a special class of entire functions and on related extremal properties of entire functions of finite degree, Izv . Akad . Nau k SSS R 1 4 (1950) , no. 1 , 45-84. (Russian ) 81. , Generalization of a theorem of Cartwright concerning an entire function of finite degree bounded on a sequence of points, Izv . Akad. Nauk SSS R 21 (1957) , 549-558. (Russian ) 82. , Distribution of zeros of entire functions, Transl . Math . Monographs , vol . 5 , Amer . Math. Soc , Providence , RI , 1980 . 83. B . Ya . Levi n an d Di n Tha n Hoa , Interference operators on entire functions of exponential type, Funktsional . Anal , i Prilozhen . 3 (1969) , no . 1 , 48-61 ; Englis h transl . i n Functiona l Anal. Appl . 3 (1969) , 39-50 . 84. N . Levinson , Gap and density theorems, Amer . Math . Soc . Colloq . Publ. , vol . 26 , Amer . Math. Soc , Ne w York , 1940 . 85. W . A . J . Luxemburg , On an inequality of Levinson of the Phragmen-Lindelof type, Indag . Math. 46 (1984) , 421-427 . 86. Yu . I . Lyubarskii , Properties of systems of linear combinations of powers, Leningra d Math . J. 1 (1990) , no. 6 , 1297-1370 . 87. Yu . I . Lyubic h an d V . A . Tkachenko , A new proof of the fundamental theorem on functions periodic in the mean, Teor . Funktsi i Funktsional . Anal , i Prilozhen . 1967 , no . 4 , 162-170 . (Russian) 88. , The abstract quasianalyticity problem, Teor . Funktsi i Funktsional . Anal , i Prilozhen. 1972, no . 16 , 18-29. (Russian ) 89. P . Malliavin, Sur la croissance radiale d'une fonction meromorphe, Illinoi s J. Math. 1 (1957) , 259-296. 90. P . Malliavin and L . A. Rubel, On small entire functions of exponential type with given zeros, Bull. Soc . Math. Franc e 89 (1961) , 175-206 . 91. S . Mandelbrojt , Series adherentes, regularisation des suites, applications, Gauthier-Villars , Paris, 1952 . 92. , Sur une probleme de Gelfand et Shilov, Ann . Sci . Ecol e Norm . Sup . (3 ) 7 7 (1960) , no. 2 , 145-166 . 93. , Theorems of closure and theorems of composition, Inostr . Liter. , Moscow , 1962 . (Russian) 94. A . S . Markus , Introduction to the spectral theory of polynomial operator pencils, Transl . Math. Monographs , vol . 71 , Amer. Math . Soc , Providence , RI , 1988 . 95. V . I . Matsae v an d E . Z . Mogulskii , A division theorem for analytic functions with a given majorant, and some of its applications, Zap . Nauchn . Sem . Leningrad . Otdel . Mat . Inst . Steklov. (LOMI ) 5 6 (1976) , 73-89; English transl . i n J . Sovie t Math . 1 4 (1980) . 96. V . I . Matsaev , On the growth of entire functions that admit a certain estimate from below, Dokl. Akad. Nauk SSS R 13 2 (1960) , 283-286; English transl. in Sovie t Math. Dokl . 1 (1960) . 97. , Volterra operators obtained from self adjoint operators by perturbation, Dokl . Akad . Nauk SSS R 13 9 (1961) , 810-813 ; English transl . i n Sovie t Math . Dokl . 2 (1961) . 98. S.N . Mergelyan , Uniform approximations to functions of a complex variable, Uspekh i Mat . Nauk 7 (1952) , no . 2 , 31-122; English transl . i n Amer . Math . Soc . TYansl . Ser . 1 3 (1962) . 99. A . M. Minkin, Reflection of exponents and unconditional bases of exponentials, St . Petersbur g Math. J . 3 (1992) , no . 5 , 1043-1068 . 100. G . W . Morgan , A note on Fourier transform, J . Londo n Math . Soc . 9 (1935) , 187-192 . 101. F. L . Nazarov, Local estimates for exponential polynomials and their applications to inequal- ities of uncertainty principle type, St . Petersbur g Math . J . 5 (1994) , 663-717 . 102. R . Nevanlinna , Analytic functions, Springer-Verlag , Berlin , 1970 . 103. N . K . Nikol'skiT , Selected problems of the weighted approximation and of spectral analysis, Trudy Mat. Inst. Steklov. 12 0 (1974) ; English transl. in Proc Steklo v Inst. Math. 12 0 (1976) . 104. , Treatise on the shift operator. Spectral function theory, Springer-Verlag , Berlin , 1986 . 105. I . V . Ostrovskii , Generalization of the Titchmarsh convolution theorem and the complex- valued measures uniquely determined by their restrictions to a half-line, Lectur e Note s i n Mathematics, vol . 1155 , Springer-Verlag, Berlin , 1985 , pp. 256-283 . 106 , On a class of entire functions, Sovie t Math . Dokl . 1 7 (1976) , 977-981 . 107. I . V . Ostrovski i an d A . M . Ulanovskii , Classes of complex-valued Borel measures uniquely determined by their restrictions, Zap . Nauchn . Sem . Leningrad . Otdel . Mat . Inst . Steklov . (LOMI) 17 0 (1989) , 233-253 ; English transl . i n J . Sovie t Math . 6 3 (1993) . BIBLIOGRAPHY 243

108. V. P . Palamodov , Generalized functions and harmonic analysis, Itog i Nauk i i Tekhniki : Sovremennye Problem y Mat. : Fundamental'ny e Napravleniya , vol . 72 , VINITI , Moscow , 1991, pp . 5-134 ; Englis h transl . i n Encyclopaedi a o f Math . Sci. , vol . 72 , Springer-Verlag , Berlin, 1995 . 109. R . E . A . C . Pale y an d N . Wiener, Fourier transforms in the complex domain, Amer . Math . Soc, Ne w York, 1934 . 110. A . Planchere l an d G . Polya , Fonctions entieres et integrates de Fourier multiples II , Com - ment. Math . Helv . 1 0 (1936) , 110-163 . 111. G . Poly a and G . Szego , Problems and theorems in analysis. I , Springer-Verlag , Berlin , 1972 . 112. I . I . Privalov , Randeigenschaften analytischer Funktionen, VE B Deutsche r Verla g Wiss. , Berlin, 1956 . 113. A. Yu. Rashkovskii, Majorants of harmonic measures and uniform boundedness of a family of subharmonic functions, Analyti c Method s i n the Probabilit y Theor y an d Operato r Theory , "Naukova Dumka" , Kiev , 1990 , pp. 115-126 . (Russian ) 114. , On the radial projection of a harmonic measure, Operato r Theory , Subharmoni c Functions, "Naukov a Dumka" , Kiev , 1991 , pp. 95-102 . (Russian ) 115. R . M . Redheffer , Completeness of sets of complex exponentials, Adv . i n Math . 2 4 (1977) , 1-62. 116. L . I. Ronkin, Introduction in the theory of entire functions of several variables, Transl . Math . Monographs, vol . 44, Amer . Math . Soc , Providence , RI , 1974 . 117. , Functions of completely regular growth, Kluwer , Dordrecht , 1992 . 118. W. Rudin , Functional analysis, McGraw-Hill , Ne w York , 1973 . 119. L . Schwartz , Theorie generate de fonctions moyenne-periodiques, Ann . o f Math . (2 ) 4 8 (1947), 857-929 . 120. K . Seip , On the connection between exponential bases and certain related sequences in L2(-7r,7r), J . Funct . Anal . 13 0 (1995) , 131-160 . 121. A . A . Shkalikov , A system of functions, Mat . Zametk i 1 8 (1975) , no . 6 , 855-860 ; Englis h transl. i n Math. Note s 1 8 (1975) , 1097-1100 . 122. A . Ph . Timan , Theory of approximation of functions of a real variable, MacMillan , Ne w York, 1963 . 123. M . Tsuji , Potential theory in modern function theory, reprint , Chelsea , Ne w York , 1975 . 124. R . S . Yulmukhametov, Approximation of subharmonic functions, Analysi s Math . 1 1 (1985) , no. 3 , 257-283. This page intentionally left blank Author Inde x

Agmon S. , 16 1 Goldberg A. A. , 13 , 57, 118 , 149 , 18 8 Ahlfors L. , 78 , 19 1 Golovin V. D. , 17 0 Akhiezer N . L , 162 , 189 , 227, 230, 235, 236 Gorin E . A. , 41, 78, 23 8 Avetisyan A . E. , 5 1 Govorov N. V., 5 7 Azarin V. S. , 78 , 96, 113 , 11 9 Grabb M . T., 23 6 Grishin A . Ph. , 57 , 113 , 14 9 Babenko K . L , 200 , 204 , 20 6 Gurarii V . P. , 18 3 Ber G . Z. , 159 , 16 2 Berenstein C, 12 4 Hormander L. , 19 7 Bernstein S . N., 150 , 160 , 162 , 227, 23 6 Hadamard J. , 18 , 26, 31, 48, 10 5 Bernstein V. , 5 5 Hardy G . H. , 50 , 6 0 Beurling A., 40, 122 , 132 , 133, 195, 197, 21 3 Havin V . P. , 19 5 Bieberbach L. , 7 3 Hayman W . K. , 13 , 40, 45 , 49, 57 , 97 , 109 , Blaschke W., 10 4 112, 11 3 Boas R . P. , 160 , 16 2 Herglotz G. , 9 9 Bonsall F . F. , 236 , 23 8 Hermite Ch. , 22 2 Borel E. , 18 , 28, 30, 6 5 Higgins J. R. , 15 1 Borichev A . A. , 11 9 Hruscev S . V., 17 8 Bourbaki N. , 4 1 Browder A. , 23 6 Ingham A. , 17 7 Brudnyi Yu . A. , 4 1 Joricke B. } 19 5 Caratheodory C, 7 5 Jensen J . L . W. V. , 9 , 10 , 4 8 Carleman T., 105 , 107 , 187 , 209, 21 3 Kadets M . I. , 17 2 Carlson F. , 58 , 71, 189 Kahane J.-P. , 12 4 Cartan H. , 12 , 76, 7 8 Katsnelson V . E. , 172 , 23 6 Cartwright M . L. , 127 , 16 0 Katznelson Y., 20 3 Chebotarev N . G. , 221 , 222, 22 5 Keldysh M . V. , 18 1 Davydova E. , 16 1 Kennedy P . B. , 45, 4 9 de Brange s L. , 130 , 22 0 Khabibullin B . N. , 85, 13 0 Din Than Hoa , 16 2 Kheifits A . I. , 16 8 Domar Y. , 11 9 Khurgin Ya . L , 15 1 DuffinR. J. , 23 0 Kjellberg B. , 4 0 Duncan T\ , 23 8 Koldobskii A . L. , 7 8 Dzhavadov M . G. , 18 3 Koosis P., 16 , 130 , 133 , 135 , 141, 145, 14 6 Dzhrbashyan M . M. , 51 , 166, 195 , 204, 20 6 Korevaar J. , 13 5 Kostyuchenko A . G. , 18 3 Eremenko A . E. , 16 5 Kotel'nikov V. , 15 1 Essen M. , 40 , 113 , 14 4 Krasichkov-Ternovskii I . F. , 133 , 21 2 Evgrafov M . A. , 4 0 Krein M . G. , 44 , 115 , 118 , 121 , 169 , 171 , 184, 215 , 218, 220, 22 2 Garnett J . B. , 141 , 145, 146 , 17 9 Gasymov M . G. , 18 3 Laguerre E. , 28 , 22 0 Gelfand I . M. , 41 , 42, 44, 119 , 199 , 204 , 23 7 Landkof N . S. , 49 , 7 8 Gelfond A . O. , 2 2 Le Page C. , 41 , 43 Gohberg I . Z. , 121 , 169 , 171 , 184, 21 5 Leau L. , 7 2

245 246 AUTHOR INDE X

Leont'ev A . F., 84 , 14 9 Plancherel M. , 50 , 152 , 16 1 Levin B . Ya. , 16 , 55 , 78 , 96 , 149 , 161 , 162 , Privalov I . L , 14 6 168, 220 , 225, 23 6 Rafaelyan S . G. , 16 6 Levinson N. , 16 , 127 , 135 , 21 3 Raikov D . A. , 41, 44 Lindelof E. , 1 , 33, 37, 55, 9 3 Rashkovskii A . Yu. , 21 3 Logvinenko V. , 16 1 Redheffer R . M. , 135 , 20 3 Luxemburg W. A . J. , 5 1 Riesz F., 48 , 9 9 Lyubarskii Yu . I. , 18 4 Riesz M. , 142 , 16 9 Lyubich Yu . I. , 107 , 12 2 Ronkin L . I. , 45 , 49, 9 6 Rubel L. , 8 5 Miintz Ch . H. , 10 3 Rudin W. , 41 , 44 Malliavin P. , 85, 132 , 133 , 16 1 Russakovskii A . M. , 14 9 Mandelbrojt Sh. , 107 , 203, 204, 20 6 Markus A . S. , 18 4 Schaeffer A . C. , 23 0 Markushevich A . I. , 2 2 Schwartz L. , 121 , 122, 12 4 Matsaev V . I. , 118 , 209, 212, 21 4 Seip Kr., 17 9 Meiman N . N., 222 , 22 5 Shannon C . S. , 15 1 Mergelyan S . N. , 8 5 Shilov G . E. , 41, 44, 58, 199 , 20 4 Minkin A . M. , 17 9 Shkalikov A . A. , 183 , 18 4 Mogulskii E. Z. , 21 2 Sinclair A . M. , 23 6 Montel P. , 221, 222 Sodin M . L. , 16 5 Morgan G . W. , 59 , 19 7 Szego G. , 4 0 Muckenhoupt B. , 17 9 Taylor B . A. , 12 4 Nazarov F. L. , 61 , 19 5 Timan A . F. , 22 7

Nevanlinna F. } 189 , 19 0 Titchmarsh E . C. , 11 9 Nevanlinna R. , 10 , 12 , 13 , 78, 99 , 105 , 187 , Tkachenko V. A. , 107 , 12 2 189, 190 , 192 , 20 9 Tsuji M. , 4 0 Nikol'skii N. K. , 121 , 178, 21 2 Ulanovskii A . M. , 11 9 Ostrovskii I . V., 13 , 57, 118 , 119 , 149 , 18 8 Ostrowski A . M. , 105 , 20 4 Valiron G. , 35 , 91, 93

Polya G. , 15 , 40, 50 , 63, 132 , 152 , 161 , 220 Weierstrass K. T . W. , 2 5 Palamodov V . P. , 20 4 Wiener N. , 69 , 133 , 146 , 17 2 Paley R . E . A . C, 69 , 133 , 146 , 17 2 Wigert S. , 7 2 Pavlov B . S. , 17 8 Yakovlev V. P. , 15 1 Phragmen E. , 1 , 3 7 Yulmukhametov R . S. , 7 8 Pichorides S . K. , 14 4 Subject Inde x

Ahlfors theorem , 19 1 Dzhrbashyan uniquenes s theorem , 19 6 Approximate identity , 13 9 Dzhrbashyan's theore m o n triviality o f clas s Babenko's theorem , 20 0 Sp, 19 9 Babenko-Dzhrbashyan theore m o n nontrivi - Element Hermitian , 23 6 ality o f clas s C(Zfc,ra n), 20 6 Entire functio n Bernstein interferenc e theorem , 16 2 admitting a lowe r bound , 20 9 Bernstein's inequality , 22 7 of completel y regula r growth , 9 4 generalized, 23 2 of exponential typ e (EFET) , 4 Beurling's theorem , 19 7 of finite order , 3 Blaschke product , 10 4 of intege r order , 3 2 Boas-Bernstein interpolatio n theorem , 16 0 of noninteger order , 3 1 Borel transform , 65 , 6 9 sine-type, 16 3 C°-set, 8 6 with zero s on a ray , 8 1 Caratheodory's inequality , 7 5 Entire functions wit h values in Banach alge - Carleman bras, 4 0 formula, 187 Exponent o f convergence , 1 7 transform, 12 2 Function Carleman's theorem , 10 5 p-trigonometric, 5 3 Carleman-Ostrowski theorem , 10 6 p-trigonometrically convex , 5 4 Carlson admitting positive harmonic majorant, 10 2 analytic continuatio n theorem , 7 1 counting, 1 0 uniqueness theorem , 5 8 for half-plane , 18 8 Cart an's estimate , 7 7 harmonic Cartwright class , 97 , 11 5 in the uppe r half-plane , 20 9 Cartwright's theorem , 16 0 positive, 10 0 Cartwright-Levinson theorem , 12 7 logarithmically subharmonic , 5 0 Class mean periodic , 12 1 (J)-quasianalytic, 1 5 subharmonic, 4 5 (A)-quasianalytic, 10 5 supporting, 6 3 C, 11 5 Functional analytic , 7 3 C(lk,mn), 20 4 P, 21 7 Gauss' theorem, 23 1 5,m, 19 9 Gelfand an d Shilo v uniqueness problem, 20 4 Completeness Gelfand proble m on invariant subspaces, 11 9 of a syste m o f exponentials , 1 9 Genus of a syste m o f exponentials, 84 , 13 2 of a canonica l product , 2 6 of a system o f functions , 2 2 of a n entir e function , 2 7 twofold, 18 1 Golovin's theorem , 17 0 Continuation analytic , 7 0 Hadamard factorizatio n theorem , 2 6 Density Hardy space , 13 7 angular, 9 5 Hardy's theorem , 6 0 lower, 1 7 Harmonic majoran t principle , 4 6 maximal, 13 2 Hay man's theorem , 10 9 upper, 1 7 Hermite-Biehler theorem , 22 2

247 248 SUBJECT INDE X

generalized, 22 2 on minimality , 13 3 Hilbert transform , 14 5 Phragmen-Lindelof theorem , 3 7 discrete analogs , 15 9 in F. an d R . Nevanlinn a form , 19 0 in integra l form , 5 0 Indicator diagram , 6 5 Plancherel-Polya conjugate, 6 5 equivalence nor m theorem, 16 1 Indicator function , 5 3 interpolation theorem , 15 2 Ingham's theorem, 17 7 Poisson formula , 9 Interference phenomenon! , 16 2 Poisson-Jensen formula , 9 Jensen formula , 10 , 4 8 Potential logarithmic , 4 8 generalized, 12 5 Product absolutel y convergent , 2 5 Katsnelson's theore m Radius spectral , 4 2 on Ries z bas e o f exponentials, 17 2 Riesz F . on spectra l radiu s o f Hermitia n element , measure, 4 8 236 theorem, 4 8 Krein simpl e fraction s serie s theorem, 11 6 Riesz M . Krein's theore m base, 16 9 on function s o f class C , 11 5 theorem, 14 2 on meromorphi c function s wit h interlac - Riesz-Herglotz formula , 9 9 ing zero s and poles , 22 0 Sampling theorem , 15 0 Laplace transform , 6 7 Schwarz formula , 9 Legendre transform, 19 5 Set o f finite view , 10 9 Lindelof's theorem , 33 , 9 3 Shilov's theorem , 5 8 Lower boun d Space for harmoni c function , 7 6 A{D), 2 0 for logarithmi c potential , 7 7 A*(D), 7 3 for th e modulu s o f a n analyti c function , H^_, 13 7 79 Hv_, 14 5 Be, 15 0 Matsaev's theorem , 209 , 21 4 L , 15 0 Maximum Principle , 3 7 a L , 14 9 Morgan's theorem , 5 9 a Spectral synthesis , 12 2 Muckenhoupt condition , 17 9 System minimal , 13 1 Muntz' theorem , 10 3 Theorem Nevanlinna on a segmen t o n th e boundar y o f th e in - characteristic, 1 2 dicator diagram , 83 , 85 class, 11 6 on additio n o f indicators, 11 8 first theorem , 1 2 on completenes s an d minimalit y o f expo - formula, 18 7 nentials, 13 4 for a half-disk , 19 2 on division , 8 0 for functions with positive imaginary parts, on tw o constant , 9 2 100 on thre e circles , 4 8 representation o f function s wit h positiv e Titchmarsh convolutio n theorem , 11 9 harmonic majorants , 10 5 Type Nevanlinna F . an d R . theorem , 18 9 exponential, 4 Operator interference , 16 2 maximal, 4 Operator preservin g inequality , 23 0 mean, 4 Order o f an entir e function , 3 minimal, 4 Ostrowski function , 20 4 normal, 4 of function , 4 P-majorant, 22 7 Polya's theore m Uncertainty Principle , 19 5 on (J)-quasianalyticity o f lacunary Fourie r Valiron's theorem, 91 , 93 series, 1 5 Weierstrass canonica l product , 2 5 on conjugat e diagram , 6 6 Weierstrass primar y factor , 2 5 Paley-Wiener Wigert-Leau theorem , 7 2 #£-theorem, 14 6 theorem, 6 9