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.