http://dx.doi.org/10.1090/ulect/033 University LECTURE Series

Volume 3 3

Interpolation an d Sampling in Space s of Analytic Function s

Kristian Sei p

American Mathematical Societ y Providence, Rhod e Islan d EDITORIAL COMMITTE E Jerry L . Bon a (Chair ) Eri c M . Friedlande r Nigel J . Hitchi n Pete r Landwebe r

2000 Mathematics Subject Classification. Primar y 30D45 , 30D50 , 30D55 , 30E05 , 42A99 , 46E15, 46E20 , 47A57 .

For additiona l informatio n an d update s o n this book , visi t www.ams.org/bookpages/ulect-33

Library o f Congress Cataloging-in-Publicatio n Dat a Seip, Kristian , 1962- Interpolation an d sampling i n spaces o f analytic function s / Kristia n Seip . p. cm . (Universit y lectur e series , ISS N 1047-3998 ; v. 33) Includes bibliographica l reference s an d index. ISBN 0-8218-3554- 8 (alk . paper ) 1. Analyti c functions . 2 . Hard y classes . 3 . Generalized spaces . 4 . Interpolation . I . Title . II. Universit y lectur e serie s (Providence , R.I.) ; 3 3 .

QA331 .S435 200 4 515'.9-dc22 2003070914

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 make fai r us e of the material, suc h a s to copy a chapter fo r use in teachin g o r research . Permissio n i s granted t o quote brie f passage s fro m thi s publicatio n i n reviews, provide d the customary acknowledgmen t o f the source i s given. Republication, systemati c copying , or multiple reproduction o f any material in 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 shoul d b e addressed to the Acquisitions Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Requests ca n also b e made b y e-mail t o [email protected] .

© 200 4 by the author. Al l rights reserved . Printed i n the United State s o f America. @ Th e paper use d i n this boo k i s acid-free an d falls withi n the guidelines established t o ensure permanenc e an d durability. Visit th e AMS home pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 9 08 07 06 05 0 4 Contents

Acknowledgements v Introduction vi i Chapter 1 . Carleson' s interpolatio n theore m 1 Chapter 2 . Interpolatin g sequence s an d the Pic k property 1 5 Chapter 3 . Interpolatio n an d samplin g i n Bergman space s 4 1 Chapter 4 . Interpolatio n i n the Bloc h spac e 6 3 Chapter 5 . Interpolation , sampling , an d Toeplit z operator s 7 9 Chapter 6 . Interpolatio n an d samplin g i n Paley-Wiener space s 9 5 Bibliography 12 5 Index 13 5 This page intentionally left blank Acknowledgements

This book i s based o n six lectures I gave in the winter o f 2003 while I was a Visitin g Professo r a t th e Universit y o f Michigan , An n Arbor , supporte d by th e Fre d an d Loi s Gehrin g Professorshi p Fund . I a m muc h indebte d t o Fred an d Loi s Gehring , fo r th e generou s suppor t throug h th e Professorshi p Fund an d fo r thei r hospitalit y durin g m y sta y i n An n Arbor . I woul d als o like to thank Fre d Gehrin g fo r hi s encouragement durin g th e preparatio n o f this book . I thank th e Departmen t o f Mathematics a t th e Universit y o f Michigan , Ann Arbo r fo r the appointment an d fo r providing me with excellen t workin g conditions. I a m particularl y gratefu l t o Pete r Dure n an d hi s wif e Ga y fo r their friendl y car e during m y sta y i n Ann Arbor . Pete r Dure n rea d throug h most o f the early drafts o f the six chapters o f this book, an d I have benefitte d from a numbe r o f correction s an d suggestion s fro m him . Artur Nicola u rea d earl y draft s o f Chapter 4 and helpe d m e understan d his beautifu l wor k wit h Bjart e E>0e . Chapter s 5 an d 6 ar e influence d b y discussions with Andreas Hartmann durin g the summer o f 2001. Discussion s with Antoni o Serr a wer e helpfu l i n workin g ou t som e part s o f Chapte r 2 , and I als o benefitte d fro m remark s fro m Donal d Marshal l concernin g tha t chapter. Th e referee s reportin g o n th e draf t submitte d t o th e AM S gav e very valuabl e feedback , an d I believ e thi s le d t o substantia l improvements . Yurii Lyubarskii' s readin g o f th e entir e manuscrip t resulte d i n a lon g lis t of pertinen t remarks . I n th e fina l stage s o f th e writing , I als o receive d a numbe r o f correction s an d constructiv e suggestion s fro m Helg e Holden , Joaquim Ortega-Cerda , an d Antoni o Serra . I than k thes e colleague s fo r their essentia l contribution s t o th e book . Finally, I than k Yuri i Lyubarski i an d Joaqui m Ortega-Cerd a fo r th e mathematics I learned through m y long-time collaboration with them. Muc h of this collaboratio n i s reflected i n the book . This page intentionally left blank Introduction

The presen t boo k i s my attemp t t o vie w the sizabl e literatur e o n inter - polating sequence s fo r space s o f analytic function s a s on e subject . I believ e the topic merits suc h a consideration, an d I think i t may benefi t fro m takin g a somewha t genera l poin t o f view . The followin g ar e the classica l origin s o f ou r subject : (1) Th e Nevanlinna-Pic k proble m wa s studie d independentl y b y Pic k

[Pil6] an d Nevanlinn a [Neval9] . Give n z\, 22 , ••• > zn an d ai, a2,..., a n i n th e unit dis k D , i t ask s fo r condition s unde r whic h th e interpolatio n proble m f(zj) — dj, j = 1,2 , ...,n , ha s a solutio n / , analyti c i n D an d \f(z)\ < 1 , z G B. Pick' s theore m i s th e following . Th e interpolatio n proble m ha s a solution i f and onl y i f the matri x

1 - a]a k \

1 - ZjZkJ j ?A:=l,...,n is positiv e semi-definite . Her e th e functio n / ca n alway s b e take n t o b e a Blaschke produc t o f degre e a t mos t n ; Nevanlinn a [Neva29 ] late r gav e a parametrization o f al l solutions . Nevanlinna-Pic k interpolatio n i n variou s guises has grown into a vast subject . Th e main reason fo r the interest i n thi s topic ove r th e las t decade s i s the recognitio n o f it s connection s t o operato r theory an d linea r system s theory . Generalization s o f i t pla y a n importan t role i n H°° contro l theory . (2) Carleson' s interpolatio n theore m [Ca58 ] give s a geometri c descrip - tion o f thos e sequence s o f point s ^1,^2,^3,.. . i n th e uni t dis k havin g th e property tha t th e interpolatio n proble m f(zj) = dj, j = 1,2,3,.. . i s solv - able b y a bounde d analyti c functio n / fo r eac h bounde d sequenc e o f dat a ai,<22,a3,.... Thi s theore m ha s playe d a distinguishe d rol e i n th e stud y o f Hardy space s fo r mor e tha n fort y years . Th e resul t appeare d firs t a s par t of a n effor t t o understan d H°° a s a Banac h algebra . I t i s intimately linke d to Carleson' s subsequen t solutio n o f th e coron a problem , an d i t als o le d naturally t o th e notio n o f a Carleso n measure , whic h late r cam e t o pla y a crucial rol e in the developmen t o f BMO. Its broad impac t i s stressed b y Pe - ter Jone s i n hi s tribute t o Carleson' s wor k i n Analysi s [Jo95] : "Thi s resul t is no w understoo d t o b e on e o f the pillar s o f functio n theory , an d i t show s up i n area s rangin g fro m th e coron a proble m t o operator theor y (an d man y places between). "

vii viii INTRODUCTIO N

(3) Th e samplin g theore m o r the Whittaker-Kotelnikov-Shannon theo - rem i n communication theor y [Sh49 ] say s that an y L 2(R) functio n / whos e Fourier transfor m /(£ ) = (2n)~ 1^2 J f(t)e~ lt^dt i s supported o n [—7r,7r] , ca n be represente d a s

The convergenc e i s both i n L 2 an d unifor m o n R , an d w e als o hav e

00 /»oo £ i^')i 2= / i/(*)i 2rf*- J=-oc J -°° The sampling theorem i s of fundamental importanc e in digital signal process- ing a s i t provide s the theoretica l foundatio n fo r digita l codin g o f continuou s waveforms an d th e decodin g o f digita l signal s (A/ D an d D/ A conversion) . *** The sequence s o f points zi, 22, £3,... describe d b y Carleson's theorem ar e called interpolating sequences. A s noted b y Shapiro and Shield s in 1961 , this notion makes sense for any reasonable space o f analytic functions. Numerou s results o n suc h sequence s hav e accumulate d sinc e that time . However , i t i s only durin g the las t decad e that result s parallelin g Carleson' s theore m hav e been foun d fo r thos e classica l space s mos t closel y relate d t o Hard y spaces , such a s Bergma n an d Dirichle t spaces . Mor e generally , thank s t o develop - ments durin g th e 1990's , th e functio n theorie s o f Bergma n an d Dirichle t spaces ar e no w o n a mor e equa l footin g wit h th e classica l H p theory . The samplin g theorem exhibit s a n exampl e o f an interpolatin g sequenc e for th e Paley-Wiene r spac e PW 2, whic h i s the spac e consistin g o f L 2 func - tions whos e Fourie r transform s ar e supporte d o n [—7r,7r] . Bu t ther e i s much mor e structure : Th e samplin g theore m show s that th e Paley-Wiene r space ha s a n orthonorma l basi s o f reproducing kernels . I n particular , ever y / G PW2 i s uniquely determine d b y it s values /(j), j G Z, an d th e L 2 nor m of / i s controlle d b y th e £ 2 nor m o f th e sequenc e o f value s f(j). I n ou r terminology, thi s mean s tha t Z i s als o a samplin g sequenc e fo r PW 2. At - tempts t o understand th e geometry o f interpolating an d samplin g sequence s for PW 2 begi n with th e wor k o f Paley an d Wiene r o n nonharmonic Fourie r series, an d ther e exist s no w a complet e an d beautifu l descriptio n o f them . The poin t o f vie w stresse d i n thi s book , tha t 'sampling ' appear s a s a dua l notion o f 'interpolation' , stem s fro m wor k b y Beurlin g i n th e lat e 1950's . Obtaining informatio n abou t th e geometr y o f interpolatin g an d samplin g sequences ca n b e viewe d a s a matter o f exploring the precis e meaning o f th e signal theoreti c phras e 'th e Nyquis t rate' . Thu s part s o f ou r stud y ca n b e given a purel y signa l theoreti c interpretation . The book i s about understandin g the geometry o f interpolating and sam - pling sequence s i n classica l Banac h space s o f analytic functions . I have ha d no intentio n t o writ e a n encyclopedia ; m y ai m ha s rathe r bee n t o describ e INTRODUCTION IX

and explai n wha t see m t o b e the mai n structura l propertie s encountere d i n the study o f such sequences . I n parts o f the book , w e take a n abstrac t poin t of vie w without specifyin g th e spac e a t hand . However , th e stag e i s mainl y occupied b y Hard y spaces , subspace s o f Hard y spaces , o r b y th e closel y related Bergma n an d Dirichle t spaces . Mos t o f th e time , w e wil l wor k i n one comple x variabl e an d i n a Hilber t spac e setting . A t firs t sight , i t ma y seem a narro w environment , bu t a s w e g o along , w e wil l se e a territor y o f considerable diversity . *** We no w giv e a n outlin e o f the content s o f the book . I n abstrac t terms , this i s mainly wha t th e boo k i s about: Le t H b e a Hilbert spac e o f analyti c functions o n som e domai n ft i n C Suppos e poin t evaluatio n / »— > f(z) i s a bounde d functiona l fo r eac h z G ft s o tha t H ha s a reproducin g kerne l

kz G H fo r eac h z G ft. Thi s mean s tha t f(z) = (/ , k z)n for eac h / G H. Let Z = (ZJ) be a sequenc e o f distinct point s fro m ft, an d conside r th e ma p 2 /i— > (f(zj)/\\k Zj ||) . Fo r which Z i s this a map into l ? Unde r the assumptio n about boundednes s fro m H t o f 2, whe n i s the ma p right-invertibl e and/o r left-invertible, o r equivalently, surjectiv e and/o r injectiv e wit h close d range ? When right-invertibilit y holds , w e sa y tha t Z i s a universa l interpolatin g sequence1 fo r H\ whe n left-invertibilit y holds , w e sa y tha t Z i s a samplin g sequence fo r H. The firs t tw o chapter s ar e reall y abou t Carleson' s theorem . Chapte r 1 presents Carleson' s theorem an d differen t approache s t o it . W e discuss Car - leson's origina l dualit y argument , th e lin k t o Carleso n measures , construc - tive proof s an d i n particula r Pete r Jones' s remarkabl y simpl e interpolatio n formula. W e sho w ho w Jones' s formul a i s linke d t o constructiv e solution s of th e d equatio n wit h L°° estimates . Th e coron a theore m i s stated , bu t not proved . W e onl y giv e a brie f indicatio n o f ho w i t i s relate d t o th e in - terpolation theore m an d solution s o f d equations . Th e chapte r end s wit h a mention o f the "Hilber t spac e approach " t o Carleson' s theorem . Chapter 2 shoul d b e viewe d mainl y a s a n investigatio n o f th e relatio n between Nevanlinna-Pic k interpolatio n an d Carleson' s theorem , bu t fro m a genera l Hilber t spac e poin t o f view . Mor e precisely , th e ai m i s t o sho w how Carleson' s theore m ca n b e viewe d a s a consequenc e o f Pick' s theorem . The chapte r ha s it s roo t i n the operato r theoreti c approac h t o Nevanlinna - Pick interpolation , pioneere d b y Sarason , an d i t i s largel y base d o n mor e recent wor k b y Agle r an d b y Marshal l an d Sundberg . Th e ide a i s to vie w the algebr a H°°(D) o f bounded analyti c function s o n the uni t dis k D a s th e multiplier algebr a o f a particula r Hilber t space , namel y H 2(D). I f Ai(H) is the multiplie r algebr a o f the Hilber t spac e H o n th e domai n ft, the n th e interpolating sequence s fo r Ai(H) ar e thos e sequence s Z = (ZJ) o f distinc t

When w e sa y tha t Z i s a n interpolatin g sequence , w e mea n tha t th e imag e o f thi s map include s £ 2. I n man y cases , ther e i s n o distinctio n betwee n interpolatin g sequence s and universa l interpolatin g sequences . X INTRODUCTION points fro m f i fo r which the interpolation proble m ip(zj) = a,j ha s a solution cp £ M(H) fo r ever y bounded sequenc e (CLJ). By Sarason' s approach , Pick' s theore m ca n b e viewe d a s a statemen t about th e multiplie r algebr a o f H 2 (D). Ther e i s a remarkably ric h clas s o f Hilbert space s H fo r whic h Pick' s theore m holds ; w e sa y that suc h space s have th e Pic k property . Thi s clas s include s iJ 2(D), th e classica l Dirichle t space, a s wel l a s so-calle d loca l Dirichle t spaces , a s show n b y Shimorin . When H ha s the Pick property, the interpolating sequence s fo r M(H) coin - cide with the universal interpolating sequences fo r H. No w Carleson's theo- s rem ca n b e viewe d a s saying that i f the ma p / H- > (f{zj)/\\kZj\\) * bounded from H 2 t o £ 2 and Z satisfie s a trivial separatio n condition , the n that ma p is also surjective. I t appears that thi s is a general phenomenon, althoug h we are not able to prove that suc h an implication holds for every H enjoyin g th e Pick property . However , a general theore m whic h i s essentially du e to B0e , shows tha t th e implicatio n hold s fo r H 2 an d th e classica l Dirichle t space . This theore m give s a striking explanatio n o f the lin k betwee n interpolatin g sequences an d Carleso n measures . Chapter 3 deal s wit h interpolatio n an d samplin g i n Bergma n spaces . The Bergma n Hilber t space s ar e example s o f Hilber t space s whic h d o no t have th e Pic k property . W e explai n wh y samplin g sequence s wer e no t en - countered unde r th e assumptio n o f th e Pic k property . Roughl y speaking , for samplin g sequence s t o exist , w e nee d th e multiplie r algebr a t o b e rela - tively smal l compare d t o th e spac e H itself . Whe n th e spac e ha s the Pic k property, the n th e multiplie r algebr a i s in a sense maximal. W e then g o on to prov e certain precis e density theorems describin g interpolatio n an d sam - pling i n Bergma n spaces . I n particular , ther e i s a critica l densit y dividin g (universal) interpolatin g sequence s fro m samplin g sequences . A n essentia l ingredient i n the proo f i s a lemma o n approximatio n o f subharmonic func - tions b y logarithm s o f modul i o f analyti c functions . W e als o poin t a t ho w this lemm a i s related t o a version o f Hormander's theore m o n weighte d L 2 estimates fo r solution s o f d equations . Th e proo f give n i n Chapte r 3 leans on work by Berndtsson and Ortega-Cerda and differ s fro m m y original proof from 1993 . Th e notio n o f density use d i n this chapte r i s a counterpart o f a density use d b y Beurlin g o n th e rea l line . W e wil l sometime s refe r t o thi s kind o f density a s a Beurling density . Chapter 4 is about interpolatio n i n the Bloch space, following a paper b y B0e and Nicolau. W e begin the chapter by a fairly extensive discussion about how interpolating sequence s shoul d reflec t th e functio n theoreti c propertie s of the spac e a t hand . Thi s discussio n reveal s the limitation s o f the genera l approach pu t forwar d i n Chapter s 2-3 . I n particular , w e giv e a definitio n of interpolating sequence s rather specifi c t o the Bloc h space, reflecting tha t the Bloc h spac e i s a Lipschit z space . I n thi s way , th e descriptio n o f inter - polating sequence s yield s precise information abou t th e "rigidity " pose d o n this Lipschit z spac e b y the analyticit y o f its elements. W e find tha t th e de- scription obtaine d i n this chapte r ca n be related directl y to the descriptio n INTRODUCTION XI of interpolating sequence s fo r Dirichle t typ e space s i n Chapte r 2 . However , there seem s to b e n o simpl e relation betwee n th e tw o kind s o f interpolatio n problems. Th e proo f i n Chapter 4 relies very much on the functio n theor y o f the Bloc h space . Th e probabilisti c interpretatio n o f the Bloc h spac e enter s the proof , an d w e mak e us e o f Jones' s constructiv e solutio n o f d equation s from Chapte r 1 . The tw o fina l chapter s ar e essentiall y abou t th e Paley-Wiene r space . In Chapte r 5 , w e interpre t th e Paley-Wiene r spac e a s a subspac e o f H 2 invariant wit h respec t t o th e backwar d shift . Thi s mean s tha t i t ha s th e form Kj — H2 0 ZiJ 2, wit h 1 a n inne r function . W e the n discus s interpo - lation an d samplin g i n Kj fo r X a n arbitrar y inne r function , followin g a n approach o f Nikolsk i an d relyin g o n a basi c discover y o f Pavlov . W e begi n by observing that w e have reached anothe r extrem e compared t o the Hilber t spaces havin g th e Pic k property : No w th e multiplie r algebr a i s trivial, i.e. , it consist s onl y o f constan t functions . W e fin d tha t w e nee d th e multiplie r algebra t o b e trivia l t o hav e sequence s bein g simultaneousl y interpolatin g and samplin g sequences . A descriptio n o f interpolatio n an d samplin g i s then give n i n term s o f invertibilit y criteri a fo r Toeplit z operators . Her e the Helson-Szeg o conditio n appear s vi a th e Widom-Devinat z theore m o n invert ibility o f Toeplitz operators . Th e chapte r end s wit h a mention o f sev - eral long-standin g problem s relate d t o ou r descriptio n o f interpolatio n an d sampling., Chapter 6 is more specificall y abou t th e Paley-Wiener space . Th e chap - ter begin s wit h simpl e proof s o f basi c results , suc h a s the Plancherel-Poly a inequality, th e samplin g theorem , an d th e Paley-Wiene r theorem . W e the n state the theorem describing interpolation an d sampling in the Paley-Wiene r space. A simpl e versio n o f th e proo f i s give n fo r th e specia l cas e o f point s located i n a half-plane . Thi s proo f essentiall y reduce s t o th e on e give n i n Chapter 5 . Th e genera l cas e require s som e additiona l technicalities , bu t i s included fo r th e sak e o f completeness . I t i s then show n ho w th e condition s of th e theore m ca n b e applied . W e deduc e Kadets' s 1/ 4 theorem , an d a basic inequalit y o f Henr y Landau , vi a th e Helson-Szeg o conditio n an d th e John-Nirenberg theorem . W e als o sho w ho w the condition s lea d t o approx - imation problem s simila r t o the on e considere d i n Chapte r 3 , and ho w suc h problems ca n b e deal t wit h t o gai n informatio n abou t sequence s "a t th e Nyquist density" . What w e cal l the Paley-Wiene r spac e i s reall y th e simples t spac e i n a wide class o f Paley-Wiener spaces , define d a s follows . Le t S b e an arbitrar y measurable subse t o f R n, an d le t PW 2(S) denot e th e spac e o f function s whose Fourie r transform s ar e supporte d o n S. I n Chapte r 6 , w e mentio n the mos t basi c resul t abou t interpolatio n an d samplin g i n thi s genera l set - ting, namel y th e necessar y densit y condition s o f Henry Landa u i n term s o f Beurling densities . W e presen t th e cor e o f Landau's argument . I n particu - lar, thi s give s a mor e genera l versio n o f th e inequalit y obtaine d b y mean s of the John-Nirenber g theorem . W e mentio n som e othe r result s an d poin t Xll INTRODUCTION at possibl e direction s fo r futur e research . Ther e ar e a number o f interestin g and difficul t problem s relate d to interpolation an d samplin g i n more genera l Paley-Wiener spaces , bot h i n on e an d severa l variables . An oversimplified bu t rather illuminating way of summarizing the "Hilber t space part " o f the book , Chapter s 2- 3 and 5-6 , i s a s follows : • Unde r th e presenc e o f the Pic k property , th e geometr y o f interpo - lating sequence s depend s o n th e geometr y o f Carleso n measures ; no samplin g sequence s exist . • Whe n th e multiplie r algebr a i s "small " bu t nontrivial , Beurlin g densities divid e interpolatin g sequence s fro m samplin g sequences . • I f the multiplie r algebr a i s trivial, i t ma y occu r tha t a sequenc e i s both a n interpolatin g sequenc e an d a samplin g sequence . The n what matters , beside s Beurlin g densities , i s t o wha t exten t se - quences deviat e fro m certai n regula r sequence s (a s Z i n the cas e o f PW2), wit h th e deviatio n measure d i n term s o f the Helson-Szeg o condition. *** I hav e trie d t o mak e th e boo k essentiall y self-contained . I assum e th e reader i s familiar with the basics o f H p theor y a s found i n any o f the standar d books by Duren [Du70] , Garnett [Gar81] , or Koosis [Koo98]. I also assume the reader know s about BMO , includin g the H 1-BMO duality , whic h i s used at on e place . Chapter s I-I V an d V I o f Garnett's boo k shoul d giv e sufficien t background, i n addition t o basi c knowledge o f complex an d functiona l anal - ysis. Wit h fe w exceptions , al l results state d ar e give n self-containe d proofs . The mos t significan t exceptio n i n additio n t o Carleson' s coron a theore m i s found i n Chapte r 2 , wher e I decide d no t t o includ e a proo f o f Parrott' s lemma. Thi s i s a n operato r theoreti c resul t whic h enter s th e proo f tha t a variety o f Hilbert space s hav e the Pic k property . Parrott' s lemm a i s closel y related t o th e celebrate d Sz.-Nagy-Foia s commutan t liftin g theorem . M y decision reflect s tha t Chapte r 2 touches a n operator-theoreti c fiel d studie d in dept h i n a recen t monograp h b y Agle r an d McCarth y [AgMc02] . Par - rott's lemm a an d muc h mor e relate d t o th e Pic k propert y ca n b e foun d i n that book . The si x chapters ar e sectione d int o what I refe r t o a s paragraphs, num - bered 0,1, 2,.. in each chapter. Th e reader wil l probably recogniz e the origi n of the tex t a s a se t o f lecture notes , with th e paragraph s makin g u p a blen d of informal explanation s wit h technica l details . Eac h paragrap h i s intende d to be a logical unit; paragraph s ar e rarely mor e than tw o pages. I found thi s a usefu l wa y o f organizin g th e materia l whe n preparin g th e lectures , an d I hope i t ma y hel p th e reade r t o fin d hi s o r he r wa y throug h th e si x storie s that I try t o tell . Bibliography

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Page numbers that appea r i n boldface poin t to the page o n which the term i s defined .

Bp, 4 1 £(X), 11 8 D+(X) (onR) , 8 2 £(Z) , 8 0 D+(X) (o n M n), 11 8 M(H), 1 6 D+(Z) (o n B>) , 56 d equation , 8 D~(X) (onR) , 8 2 ~(X) (onR n), 11 8 ~ , 1 7 D_(Z) (onD) , 5 6 0(*,C) , 2 2

DQ, 1 7 £H(Z,C) , 2 1 Dj(e), 4 2 CH(Z) , 7 if, 1 5 cj(Z) , 7 tfi,83 CHJ(Z), 8 Hl8S <*(*,C) , 64

7700 x dm , 1 8 HP(C+), 1 h, 1 6 ifp(D), 1 *^ > 2 4 ify, 8 7 ™(*) > 12 #?, 8 3 Adams, D.R. , 2 0 #?, 9 0 Agler property , 26 , 28-3 1 M(Z), 2 v M '* Agler , J. , ix , xii, 13-15 , 21, 26-30 Aleksandrov, A. , 9 1 PW(5'), 11 8 Amar, E. , 12 , 43, 62 PW2, 7 9 pr, s i analytic Beso v space, 3 3 analytic Sobole v space , 3 3 p+, 1 2 analyzing wavelet , 4 5 P/, 8 5 Anderson, J . M. , 6 3 ®W> 4 Arcozzi , N. , 18 , 21, 37, 3 8 S* 8 3 5555 5 T^,' arithmetic-geometri8 6 Avdonin, S . A. , 81, c10 mea9 n inequality, 5 9 XW' 11 3 Axler , S. , 13 , 21 Z\ 10 1 B, 6 3 backwar d shift , 8 3 C+, 1 Bari' s theorem , 23 , 35 Capa, 1 9 Berenstein , C . A. , 1 4 ID5, 1 Bergma n space , 15 , 62 O, 1 5 Bergma n L p space , 32 , 61, 63 T, 1 o n bounded symmetri c domains , 4 3 <5(Z), 3 weighted , 17 , 39, 41, 42, 45, 81 <, 1 7 Berndtsson , B. , x, 13 , 56, 60, 6 1

135 136 INDEX

Bernstein space , 5 6 Cohn's conjecture , 90 , 9 1 Bernstein-Boas formula , 11 3 Cohn's theore m Bessel capacity , 1 9 on Carleso n measures , 9 0 Beurling densities, see also uniform den - on interpolation i n D a, 3 7 sity Cohn, W. , 33 , 90 Beurling, A. , x , 56 , 82, 83, 12 3 Coifman, R . R. , 4 3 Beurling, P. , 6 complete interpolatin g sequence , 7 9 Bishop, C, 15 , 32, 3 3 for K\, 89 , 9 2 Bishop-Marshall-Sundberg theorem , 3 7 for PW 2, 80 , 108 , 109 , 113 , 11 4 Bloch martingale , 6 9 for PW 2(S), 118 , 122 , 12 3 Bloch space , 15 , 38, 63, 64 , 66, 68, 7 2 for PW P, 8 2 conformal invariance , 63 , 64 , 66 , 67 , for PW P(S), 12 4 70, 7 1 complete Pic k property , 2 6 mean valu e property , 6 8 conjugation operator , 87 , 9 9 BMO, 13 , 64, 77 , 93, 100 , 107 , 11 0 constant o f interpolation, 2 , 4 9 BMOA, 64 , 7 7 Bonsall, F. F. , 1 4 Daubechies, L , 4 7 Borichev, A. , 6 2 decomposition theorem , 4 7 Bourgain, J. , 8 9 dense de Branges , L. , 8 0 £-dense, 4 2 Bruna, J. , 14 , 61 Devinatz, A. , 8 7 Buck, R . C, 1 3 Dirichlet shift , 3 0 B0e's theorem , 3 6 Dirichlet space , 15 , 17, 30, 33, 37, 64, 65 B0e, B., v, x, 2 , 33, 34, 37, 63-66, 76 , 7 7 distance ^-distance, 21 , 47 capacitary stron g type inequality , 2 0 (j-distance, 9 7 Carleson Douglas, R . G. , 89 , 9 0 condition, 3 , 4 , 5 , 12 , 31 , 32 , 36 , 68 , Drasin, D. , 5 2 75, 85 , 88, 98, 103 , 104 , 11 1 Duffin, R . J. , 47 , 8 1 for H, 3 2 Duren, G. , v for Bp, 6 1 Duren, P. , v , xii, 3 , 14 , 41, 42, 54, 6 4 for Di, 3 2 dyadic subdivision , 3 8 for D , 3 3 a Earl, J . P. , 6 for H, 32-3 4 measure, 5 , 9-11, 15 , 73-7 6 Federer, PL , 7 4 X-Carleson measure , 9 0 Fefferman's X-Carleson measure , 9 1 duality theorem , 10 7 for £/3 , 41, 42, 4 3 theorem on multipliers on the ball, 12 4 for D a, 1 8 Fefferman, C , 13 , 12 4 for H, 16 , 43, 44, 47-4 9 Fefferman-Stein decomposition , 1 3 2 for PW , 9 8 Flornes, K. , 8 2 Carleson's frame, 45 , 46 , 47, 80, 8 1 corona theorem , ix , xii, 1 0 of complex exponentials , 80 , 8 1 embedding theorem , 4 , 5 , 107 , 10 8 of normalized reproducin g kernels , 46, interpolation theorem , vii , ix , x, 3, 1 - 86 15, 6 1 of wavelets, 4 5 Carleson, L. , vii, ix , 2-6, 10-13 , 25 , 31 free interpolation , 1 5 champagne subdomain , 6 1 Frostman's theorem , 91 , 92, 100 , 10 2 Chang, S.-Y . A, 7 0 Fuglede's conjecture , 80 , 123 , 12 4 Christ, M. , 6 1 Fuglede, B. , 12 3 Cima, J . A. , 8 4 functional o f point evaluation , 12 , 15 , 16 Clark, D. , 8 0 Clunie, J. , 6 3 Gabor transform , 4 5 INDEX 137

Gamelin, T . W. , 1 1 for PW 2, 80 , 98 , 99 , 103-105 , 107 , Garnett's theore m o n harmoni c interpo - 111, 11 2 lation, 1 4 for PW V, 8 2 Garnett, J . B. , xii, 1 , 2 , 6 , 8 , 11 , 14 , 61, invariant are a measure , 5 0 76, 89 , 91, 92, 105 , 11 0 Iosevich, A. , 12 4 Gehring, F . W., v , 6 1 Ivanov, S . A., 8 1 Gehring, L. , v Girnyk, M . A. , 5 2 Jayant, N . S. , 8 0 Gorin, E . A. , 8 Jensen's formula , 5 9 Gorkin, P. , 1 3 Jevtic, M. , 6 1 Grammian, 3 4 John, F. , 11 0 Grochenig, K. , 12 0 John-Nirenberg theorem, xi, 95, 110, 120 Jones, P . W. , vii , ix , 6 , 8 , 9 , 11 , 13 , 33, Hankel operator, 8 7 61, 76 , 9 2 Hardy space , 1 , 15 , 45, 81 Kothe-Toeplitz theorem , 2 3 Hardy, G . H. , 2 9 Kadets's 1/ 4 theorem , xi , 108 , 11 4 Hardy-Littlewod maxima l theorem , 2 0 Kadets, I . M. , 10 8 harmonic measure , 6 1 Katz, N . H. , 12 4 Hartmann, A. , v , 14 , 93 Kerman, R. , 1 8 Hastings, W. W. , 4 2 Koosis, P. , xii, 2 , 8 , 3 6 Havin, V . P, 8 , 9 2 Korenblum, B. , 41, 61, 62 Hayman, W . K. , 1 3 Koszul complex , 1 1 Hedberg, L . I., 2 0 Hedenmalm, H. , 41, 62 Laba, L , 12 4 Helson, H. , 10 5 Landau, H . J. , xi , 95, 109 , 111 , 118, 12 0 Helson-Szego Levin, B . Ya., 81 , 96 condition, 87 , 88 , 99, 100 , 105 , 10 6 Lindholm, N. , 12 0 theorem, 105 , 10 6 linear operato r o f interpolation, 6 Herz, C . S. , 12 2 Lipschitz space , x , 64 , 6 5 Higgins, J . R. , 8 0 local Dirichle t space , 3 0 Hilbert transform , 10 5 logarithmic capacity , 1 9 weighted nor m estimates , 91 , 10 5 logarithmically regular partition, 114 , 11 6 Hjelle, G . A. , 8 9 Luecking, D. , 42 , 6 1 Holden, H. , v Lyubarskii, Yu. , v , 52 , 61, 105, 114 , 122 , Hormander, L. , 8 , 11 , 122 123 Hormander's theorem on weighted L 2 es - timates, 6 0 Makarov, N. , 63 , 69, 7 0 Horowitz, C, 25 , 44 Malinnikova, E. , 52 , 11 4 Hruscev, S . V., 8 , 81, 84, 93, 99, 10 9 Marco, N. , 52 , 6 1 hyperbolic Marshall, D . E., ix , 2 , 12 , 13 , 15, 29, 30, geodesic, 49 , 58, 59, 64, 67 , 10 8 32, 33 , 89, 10 1 metric, 6 4 Marshall, D . E„ v Massaneda, X. , 14 , 52 , 6 1 Ingham, A . E. , 10 9 maximal functio n interpolating Hardy-Littlewood, 2 0 Blaschke product, 85 , 89 , 101 , 10 3 nontangential, 5 , 2 0 sequence Maz'ya, V . G. , 2 0 for B, 65 , 7 7 McCarthy, J . E. , xii , 13-15 , 21, 26-30 for M(H), 2 1 McCullough, S. , 2 6 for H, 21 , 47 McCullough-Quiggin theorem , 2 6 for #°°, 2 , 5 , 11 , 13 Menini, C, 6 2 2 for K Bl 8 9 Minkin, A . M. , 9 9 for K\, 85 , 86, 8 8 Mortini, R. , 1 3 138 INDEX

Muckenhoupt's (A p) condition , 9 3 Remling, C. , 8 0 multiplication operator, 1 6 reproducing kernel , 1 6 multiplier algebra , ix , 13 , 16, 18 , 25, 44, Richter, S. , 30 , 6 2 79 Riesz basis Nakazi's conjecture , 9 3 of complex exponentials, 80 , 12 4 Nakazi, T. , 9 3 of normalize d reproducin g kernels , Narita, J. , 1 4 85 Nazarov, F. , 90 , 9 1 measure, 50-52 , 8 2 Nehari's theorem, 8 7 projection, 12 , 2 0 Nevanlinna's theorem, 2 sequence, 23 , 8 0 Nevanlinna, R. , vii , 2 , 6 of complex exponentials, 80 , 8 1 Nevanlinna-Pick problem, 1 , 2, 6, 12 , 24 Rochberg, R. , 17 , 18 , 21, 37, 38 , 43, 47, Neville, C . W., 4 , 5 93 Newman, D . J. , 1 3 Ross, W. T. , 8 4 Nicolau, A. , v , x , 8 , 14 , 63-66 , 76 , 77 , Rudin, W. , 86 , 8 9 89, 9 2 Nikolski, N . K., xi , 11 , 13-15, 23, 35, 81, sampling 83, 84 , 87, 92 , 93, 99 constant, 4 8 Nirenberg, L. , 11 0 measure, 6 2 Nollas, P. , 8 0 sequence, 62 , 79, 81 Nyquist for B, 66 , 6 7 density, 4 5 for B(3, 43, 45, 53, 56, 5 9 rate, 11 8 for H, 44 , 45, 47, 48, 86 Nyquist, H. , 4 5 for K%, 8 9 for K}, 83 , 85, 86, 88, 92 Ortega-Cerda, J. , v , x , 8 , 52 , 56 , 60 , 61, for PW 2, 80 , 97 , 99 , 104 , 105 , 107 , 66, 82 , 99, 111 , 113, 114 , 12 4 108, 110-112 , 114 , 117 , 120 , 12 1 for PW 2(S), 118-120 , 122 , 12 3 Palamodov, V . P. , 12 3 for PW P, 8 2 Paley, R . E . A . C, 81 , 108 theorem, viii , xi , 80 , 82 , 96 , 97 , 108 , Paley-Wiener 120 space, 79 , 81 , 95, 97, 118 , 12 2 Sarason, D., ix , x, 2 6 theorem, xi , 79 , 80, 96, 97, 12 2 Sawyer, E. , 18 , 21, 37, 3 8 Parrott's lemma , xii , 2 8 Schaeffer, A . C, 47 , 8 1 Pascuas, D. , 1 4 Schur product, 2 8 Pavlov, B. S. , xi, 81, 84, 93, 99 Schuster, A . P. , 32 , 41, 42, 54, 61, 64 Phragmen-Lindelof principle , 82 , 9 5 Segal, I. , 12 4 Pick property , 13 , 15, 24, 25 , 26, 29-34, Seip, K. , 8 , 25, 32, 52, 54, 56, 61, 62, 82, 36, 37 , 44, 7 9 93, 99 , 105 , 111, 113, 114, 12 2 Pick's theorem, ix , x, 1 , 2 , 1 3 separated sequence , 5 Pick, G. , vii , 1 H-separated sequence , 21, 47 Plancherel, A. , 9 6 cr-separated sequence, 98, 104 , 105, 111 Plancherel-Polya inequality , 96, 9 8 Serra, A. , v , 26 , 27 , 3 0 Polya, G. , 9 6 Shannon, C. , vii i Pommerenke, Ch. , 6 3 Shapiro, H. S., viii, 12 , 25, 29, 31, 81, 84, pseudohyperbolic metric, 2 2 90 Quiggin, P. , 26 , 2 7 Shapiro-Shields interpolation theorem , 1 3 Ramanathan, J. , 45 , 12 0 property, 2 9 Rashkovskii, A. , 12 3 Shields, A . L. , viii , 12 , 25, 29, 31, 81, 90 Razafinjatovo, H. , 12 0 Shimorin, S. , x , 3 0 reflection operator , 9 7 Spitkovsky, L , 12 2 INDEX

Stegenga's theorem , 18 , 1 9 Wolff, T. , 11 , 13, 70 Stegenga, D . A. , 18 , 37 Wu, Z. , 17 , 1 8 Steger, T. , 45 , 12 0 Stein, E . M. , 1 3 Xiao, J. , 33 , 64 Stray, A. , 14 , 89, 10 1 Young, R . M. , 8 1 Sundberg, C , ix , 2 , 12 , 13 , 15 , 29 , 30 , Yulmukhametov, R . S. , 5 2 32, 33 , 76 Sz.-Nagy-Foia§ commutan t liftin g theo - zero sequence, 2 5 rem, xii , 2 8 Zhu, K. , 41 , 47 Szego, G. , 10 5

Tao, T. , 123 , 12 4 Taylor, B . A., 1 4 Thomas, P. , 14 , 61 Thomson, J . E. , 6 2 Toeplitz operato r invertibility criterion , 87 , 93, 10 5 on H 2, 81 , 82, 86, 89 , 92 on H p, 9 3 Treil, S. , 11 , 90, 9 1 Treil-Volberg theorem , 9 0 unconditional basi c sequence , 2 3 uniform densit y lower on D , 5 6 on E , 8 2 onJET, 11 8 upper on D , 56 , 8 9 on R , 8 2 onln, 11 8 uniformly separate d sequence , 3 universal interpolatin g sequence , 12 , 1 7 for Da, 6 6 for if , 13 , 15, 79 for H 2, 1 3 for K%, 85

Varolin, D. , 6 1 Vasyunin, V . I. , 1 4 Verbitsky, I . E. , 3 3 Vinogradov, S . A., 8 , 1 3 Volberg, A. , 90 , 9 1

Wallsten, R. , 6 1 weak-operator topology , 2 4 Whittaker-Kotelnikov-Shannon theorem , see also sampling theore m Widom, H. , 8 7 Widom-Devinatz theorem, xi, 88, 92, 105 Wiener, N. , 81, 10 8 Williams, D . L. , 1 4 Wilson, J . M. , 7 0