Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2011 Paley-Wiener theorems with respect to the spectral parameter Susanna Dann Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Applied Mathematics Commons Recommended Citation Dann, Susanna, "Paley-Wiener theorems with respect to the spectral parameter" (2011). LSU Doctoral Dissertations. 3986. https://digitalcommons.lsu.edu/gradschool_dissertations/3986 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. PALEY-WIENER THEOREMS WITH RESPECT TO THE SPECTRAL PARAMETER A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Susanna Dann B.S., Fachhochschule Stuttgart, Germany, 2004 M.S., Louisiana State University, 2006 August 2011 To my family ii Acknowledgments It is a genuine pleasure to thank my advisor Professor Gestur Olafsson´ for his time, patience and guidance during my years of graduate study at Louisiana State University. I am also thankful to all my teachers, colleagues and students from and through whom I learned a lot. Special thanks go to professors Gestur Olafsson, Am- bar Sengupta, Mark Davidson, Raymond Fabec, Lawrence Smolinsky and Jacek Cygan and to my former and present fellow graduate students Natalia Ptitsyna, Amber Russell, Maria Vega, Maiia Bakhova, Jens Christensen, Silvia Jim´enezand Keng Wiboonton. I am grateful to Professor Dmitry Ryabogin from Kent State University for many discussions we had about mathematics and academia in gen- eral and for introducing me to the area of convex geometry, which became one of my research interests. During fall and summer semesters of 2005-2006 I had the opportunity to conduct research and lead undergraduate research projects in im- age processing - my research interests since my undergraduate studies - for which I am grateful to Professor Peter Wolenski, back then my fellow graduate student Stanislav Zabi´candˇ Steven Bujenovi´c,MD. from Our Lady Of the Lake PET Imaging center. My adventure of graduate studies at LSU began with the nomination as a Ful- bright scholar by the German Fulbright commission to which I am thankful for the nomination and for the financial support during my first year at LSU. Special thanks go to professors Frank Neubrander and Peter Wolenski who were instru- mental in my transition from a visiting Fulbright scholar to a graduate student in the Department of Mathematics. I am thankful to my advisor for supporting me through his NSF grant (DMS-0801010) during the summer semesters of 2008 through 2011 and for supporting my travel to several conferences abroad. During the summer semesters of 2005, 2006 and during the fall semester 2005 I was sup- ported by Professor Peter Wolenski's Board of Regents grant (107-60-4034-LEQSF) in connection with the image processing projects for which I am thankful. I would also like to thank the Department of Mathematics at Louisiana State University for supporting my numerous travels to conferences through the Louisiana Board of Regents grant (LEQSF). My special thanks go to Professor Robert Perlis who supported my travel to a workshop in Banff. The Graduate School of Louisiana State University provided me with a Graduate School Supplement Award during the years of 2006 to 2010 for which I am grateful. Last but not least, I am forever grateful to my parents, Anna and Alexander, my brother Philipp, my niece Elleonore and my better half, Olivier Limacher, for their love, support and encouragement. iii Table of Contents Acknowledgments . iii List of Symbols . vi Abstract . viii Chapter 1: Introduction . 1 Chapter 2: Paley-Wiener Theorems on Rn ........................ 3 2.1 Preliminaries . .3 2.2 Holomorphic Functions on Cn .....................5 2.3 Paley-Wiener Theorems on Rn .....................8 2.3.1 The Classical Paley-Wiener Theorem . .9 2.3.2 Paley-Wiener Theorem for L2 Functions . 14 2.3.3 Paley-Wiener Theorem for Distributions . 18 Chapter 3: Paley-Wiener Theorems for Vector Valued Functions . 21 3.1 Preliminaries . 21 3.2 Paley-Wiener Type Theorems for Vector Valued Functions on Rn . 26 H 3.2.1 The Paley-Wiener Theorem for Dr;c .............. 28 H 3.2.2 The Paley-Wiener Theorem for Dr .............. 31 3.2.3 The Special Case of H = L2(Sn−1) and SO(n)-finite Functions 35 Chapter 4: Paley-Wiener Theorems with Respect to the Spectral Parameter . 37 4.1 Preliminaries . 37 4.2 Fourier Analysis on Rn and the Euclidean Motion Group . 44 4.2.1 Gelfand Pairs . 44 4.2.2 Fourier Transform FRn Revisited . 48 4.2.3 Representations of the Euclidean Motion Group . 48 4.2.4 Fourier Transform on Rn with Respect to the Euclidean Mo- tion Group: FE(n) ........................ 49 4.3 Paley-Wiener Theorems with Respect to the Spectral Parameter . 50 n 4.3.1 Description of S(R ) with Respect to FE(n) ......... 51 n 4.3.2 First Description of Dr(R ) with Respect to FE(n) ...... 53 n 4.3.3 Second Description of Dr(R ) with Respect to FE(n) .... 57 4.3.4 Remarks . 68 Chapter 5: Extension of the Euclidean Paley-Wiener Theorem to Projective Limits . 71 5.1 Inductive and Projective Limits . 71 iv 5.2 Extension of the Classical Paley-Wiener Theorem to Projective Limits 75 5.3 Extension of the Euclidean Paley-Wiener Theorem to Projective Limits . 77 References . 82 Vita ............................................................. 86 v List of Symbols j · j Euclidean norm for a vector in Cn .................................3 (·; ·) Inner-product on Cn ...............................................3 N Set of natural numbers f1; 2;::: g .................................3 N0 Set of natural numbers including 0 . 3 j · j L1-norm for a multi-index in Nn ...................................3 Dα Partial differential operator corresponding to the multi-index α . .3 C1(Ω) Space of smooth functions on Ω . 3 1 Cc (Ω) Space of smooth compactly supported functions on Ω . 3 1 j · jK,α Seminorm on C (Ω) ...............................................3 E(Ω) Space of smooth functions equipped with the Schwartz topology . .3 supp(f) Support of a function f ............................................3 DK (Ω) Subspace of E(Ω) of functions with support in K ...................3 1 D(Ω) Cc (Ω) equipped with the Schwartz topology . 4 Br(m) Open ball of radius r > 0 centered at the point m ..................4 ¯ Br(m) Closed ball of radius r > 0 centered at the point m .................4 ¯ Dr(M) Subspace of D(M) of functions supported in Br(0) . .4 S(Rn) Space of Schwartz functions . 4 j · jN,α Schwartz seminorm . 4 f;b FRn (f) Fourier transform of f .............................................4 Sn−1 Unit sphere in Rn ..................................................4 d! Surface measure on Sn−1 ...........................................4 σn Volume of the unit sphere with respect to the surface measure . .4 µn Normalized measure on the sphere . 4 ξ(p; !) Hyperplane in Rn ..................................................4 Ξ Set of hyperplanes in Rn ...........................................4 Rf Radon transform of f ..............................................4 n−1 S(R×S ) Schwartz functions on R × Sn−1 ....................................4 n−1 D! Differential operator on S .......................................4 SH (Ξ) Certain class of Schwartz functions on Ξ . 5 ηk;m;D! (·) Seminorm on SH (Ξ) ...............................................5 DH (Ξ) Compactly supported functions in SH (Ξ) ...........................5 n−1 DH;r(Ξ) Functions in SH (Ξ) with support in [−r; r] × S ..................5 jfjm;D! Seminorm on DH;r(Ξ) ..............................................5 P (z; r) Polydisc with polyradius r and center z ............................5 ch(f) Cauchy integral of f . 6 n PWr(C ) Paley-Wiener space for smooth functions . 8 n qN;r(·) Seminorm on PWr(C ) ............................................8 n sα,r(·) Seminorm on PWr(C ) ............................................8 n n PW(C ) Inductive limit of PWr(C ) ........................................8 V 0 Continuous dual of a topological vector space V ...................18 co(E) Convex hull of the set E ..........................................22 vi O(Ω) Space of holomorphic functions on Ω . 23 j · jK Seminorm on O ...................................................23 H Dr Space of weakly Schwartz functions . 26 H νN;u(·) Seminorm on Dr .................................................26 H H Dr;c Continuous functions in Dr .......................................26 H PWr Space of weakly holomorphic functions of exponential growth ≤ r . 26 H ρN;u(·) Seminorm on PWr ...............................................26 Hl Spherical harmonics of degree l ...................................35 dn(l) Dimension of Hl ..................................................35 dn(l) fYl;igi=1 Basis for Hl ......................................................35 M(n; F) Set of square matrices of size n ...................................37 GL(n; F) General linear group . 37 exp(·) Exponential map for M(n; F) .....................................38 Gx Subgroup of G that