Volchkov Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group

1. Auflage 2009
ISBN: 978-1-84882-533-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 671 Seiten, eBook

Reihe: Springer Monographs in Mathematics

The theory of mean periodic functions is a subject which goes back to works of Littlewood, Delsarte, John and that has undergone a vigorous development in recent years. There has been much progress in a number of problems concerning local - pects of spectral analysis and spectral synthesis on homogeneous spaces. The study oftheseproblemsturnsouttobecloselyrelatedtoavarietyofquestionsinharmonic analysis, complex analysis, partial differential equations, integral geometry, appr- imation theory, and other branches of contemporary mathematics. The present book describes recent advances in this direction of research. Symmetric spaces and the Heisenberg group are an active ?eld of investigation at 2 the moment. The simplest examples of symmetric spaces, the classical 2-sphere S 2 and the hyperbolic plane H , play familiar roles in many areas in mathematics. The n Heisenberg groupH is a principal model for nilpotent groups, and results obtained n forH may suggest results that hold more generally for this important class of Lie groups. The purpose of this book is to develop harmonic analysis of mean periodic functions on the above spaces.

Zielgruppe

Research

Autoren/Hrsg.

Volchkov, Valery V.

Volchkov, Vitaly V.

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;Part I Symmetric Spaces. Harmonic Analysis on Spheres;12
3.1;Chapter 1 General Considerations;14
3.1.1;Numbers, Algebras and Groups. Some Illustrative Examples;14
3.1.2;Elements of Differential Geometry;25
3.1.3;Homogeneous and Symmetric Spaces;31
3.1.4;Convolution, Invariant Differential Operators and Spherical Functions;34
3.1.5;Structure of Quasi-Regular Representations of Compact Groups;38
3.2;Chapter 2 Analogues of the Beltrami-Klein Model for Rank One Symmetric Spaces of Noncompact Type;43
3.2.1;The Real Hyperbolic Space SO0(n,1)/SO(n);44
3.2.2;The Complex Hyperbolic Space SU(n,1)/S(U(n)xU(1));51
3.2.3;The Quaternionic Hyperbolic Space Sp(n,1)/Sp(n)xSp(1);55
3.2.4;The Cayley Hyperbolic Plane F*4/Spin(9);60
3.3;Chapter 3 Realizations of Rank One Symmetric Spaces of Compact Type;69
3.3.1;The Space SO(n+1)/SO(n);69
3.3.2;The Real Projective Space SO(n+1)/O(n);73
3.3.3;The Complex Projective Space SU(n+1)/S(U(n)xU(1));79
3.3.4;The Quaternionic Projective Space Sp(n+1)/Sp(n)xSp(1);83
3.3.5;The Cayley Projective Plane F4/Spin(9);87
3.4;Chapter 4 Realizations of the Irreducible Components of the Quasi-Regular Representation of Groups Transitive on Spheres. Invariant Subspaces;92
3.4.1;The Groups SO(n) and O(n);93
3.4.2;The Group U(n);98
3.4.3;The Group OC(n);101
3.4.4;The Group Sp(n);105
3.4.5;The Group OQ(n);128
3.4.6;The Group OCa(2);131
3.5;Chapter 5 Non-Euclidean Analogues of Plane Waves;142
3.5.1;The Case of HRn;143
3.5.2;The Case of HCn;148
3.5.3;The Case of HQn;150
3.5.4;The Case of HCa2;155
3.6;Comments, Further Results, and Open Problems;160
4;Part II Transformations with Generalized Transmutation Property Associated with Eigenfunctions Expansions;164
4.1;Chapter 6 Preliminaries;166
4.1.1;Holomorphic Functions;167
4.1.2;Distributions;176
4.2;Chapter 7 Some Special Functions;182
4.2.1;Cylindrical Functions;182
4.2.2;Jacobi Functions;185
4.2.3;Extension of the Jacobi Polynomials;193
4.2.4;Confluent Hypergeometric Functions;200
4.3;Chapter 8 Exponential Expansions;206
4.3.1;Main Classes of Distributions;207
4.3.2;Biorthogonal Systems. General Completeness Results;212
4.3.3;Expansions in Series of Exponentials;224
4.3.4;The Distribution zetaT. Solution of the Lyubich Problem;230
4.4;Chapter 9 Multidimensional Euclidean Case;235
4.4.1;Introductory Results;236
4.4.2;Spherical Functions and Their Generalizations;239
4.4.3;Hankel-Like Integral Transforms;247
4.4.4;Transmutation Operators Induced by the Converse Hankel Transform. Connection with the Dual Abel Transform;250
4.4.5;Bessel-Type Decompositions for Some Classes of Functions with Generalized Boundary Conditions;261
4.5;Chapter 10 The Case of Symmetric Spaces X=G/K of Noncompact Type;272
4.5.1;Generalities;272
4.5.2;Fourier Decompositions on G/K;277
4.5.3;Eisenstein-Harish-Chandra Integrals and Their Rank One Generalizations;281
4.5.4;The Helgason-Fourier Transform f(lambda,b);293
4.5.5;Action of f(lambda,b) on the Space E'delta(X);297
4.5.6;The Transmutation Mapping Adelta Related to the Inversion Formula for the delta-Spherical Transform;303
4.5.7;The Class E'(X) of Distributions with Radial Spherical Transform. Mean Value Characterization. Explicit Form for X=G/K (G Complex);312
4.5.8;Some Rank One Results on the Mapping Adelta;316
4.5.9;Ideas and Methods of Sect. 9.5 Applied in Analogous Problems for G/K;329
4.6;Chapter 11 The Case of Compact Symmetric Spaces;338
4.6.1;Compact Symmetric Spaces of Rank One from the Point of View of Realizations;339
4.6.2;Continuous Family of Eigenfunctions of the Laplace-Beltrami Operator;345
4.6.3;Analytic Extension of the Discrete Fourier-Jacobi Transform;356
4.6.4;The Transmutation Operators Ak,m,j Associated with the Jacobi Polynomials Expansion ;363
4.6.5;Analogues of Ak,m,j in Exterior of a Ball. The Zaraisky Theorem;368
4.7;Chapter 12 The Case of Phase Space;374
4.7.1;The Twisted Convolution of Distributions on Cn. Special Hermite Operator;375
4.7.2;Expansions over Bigraded Spherical Harmonics;377
4.7.3;Derivatives of Generalized Laguerre Functions;380
4.7.4;Analogues of the Spherical Transform;387
4.7.5;Transmutation Mappings Generated by the Laguerre Polynomials Expansion;393
4.8;Comments, Further Results, and Open Problems;398
5;Part III Mean Periodicity;403
5.1;Chapter 13 Mean Periodic Functions on Subsets of the Real Line;405
5.1.1;Main Classes of Mean Periodic Functions;406
5.1.2;Structure of Zero Sets;408
5.1.3;Nonharmonic Fourier Series;416
5.1.4;Local Analogues of the Schwartz Fundamental Principle;424
5.1.5;The Problem of Mean Periodic Continuation;427
5.1.6;One-Sided Liouville's Property;438
5.2;Chapter 14 Mean Periodic Functions on Multidimensional Domains;441
5.2.1;General Properties;442
5.2.2;Modern Versions of the John Theorem. Connections with the Spectrum. The Hemisphere Theorem;447
5.2.3;Multidimensional Analogues of the Distribution zetaT. Mean Periodic Functions with Support in Exterior of a Ball. Exactness of Uniqueness Theorems;453
5.2.4;Analogues of the Taylor and the Laurent Expansions for Mean Periodic Functions. Estimates of the Coefficients;458
5.2.5;Convergence Theorems. Extendability and Nonextendability Results;469
5.2.6;Problem on Admissible Rate of Decreasing. Reduction to the Helmholtz Equation;474
5.2.7;Hörmander-Type Approximation Theorems on Domains Without the Convexity Assumption;480
5.3;Chapter 15 Mean Periodic Functions on G/K;487
5.3.1;Preliminary Results;487
5.3.2;Uniqueness Problem. Features for Higher Ranks;492
5.3.3;Refinements for the Rank One Case;495
5.3.4;Counterexamples to the Uniqueness Problem;500
5.3.5;Generalized Spherical Functions Series;503
5.3.6;Structure Theorems and Their Applications;510
5.3.7;Sharp Growth Estimates. Comparing with Eigenfunctions of the Laplacian;515
5.4;Chapter 16 Mean Periodic Functions on Compact Symmetric Spaces of Rank One;523
5.4.1;Group and Infinitesimal Properties;524
5.4.2;Uniqueness Results;535
5.4.3;Series Development Theorems;542
5.5;Chapter 17 Mean Periodicity on Phase Space and the Heisenberg Group;545
5.5.1;Background Material;546
5.5.2;Phase Space Analogues of the Uniqueness Theorems;548
5.5.3;Characterizations of the Kernel of the Twisted Convolution Operator;554
5.6;Comments, Further Results, and Open Problems;558
6;Part IV Local Aspects of Spectral Analysis and the Exponential Representation Problem;569
6.1;Chapter 18 A New Look at the Schwartz Theory;572
6.1.1;Localization of the Schwartz Theorems. The Effect of the Size of the Domain;573
6.1.2;Pairwise Mean Periodic Functions;579
6.1.3;The Case of ``Small'' Intervals. Connections with Division-Type Problems for Entire Functions;587
6.1.4;The Deconvolution Problem. Explicit Reconstruction Formulae;590
6.2;Chapter 19 Recent Developments in the Spectral Analysis Problem for Higher Dimensions;594
6.2.1;Solution of the Berenstein-Gay Problem. Generalizations;595
6.2.2;Expansions Associated with Cylindrical Functions;601
6.2.3;More on the Berenstein-Gay Problem: The Case R=