Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group (eBook)

Preface 5
Contents 7
Part I Symmetric Spaces. Harmonic Analysis on Spheres 12
Chapter 1 General Considerations 14
Numbers, Algebras and Groups. Some Illustrative Examples 14
Elements of Differential Geometry 25
Homogeneous and Symmetric Spaces 31
Convolution, Invariant Differential Operators and Spherical Functions 34
Structure of Quasi-Regular Representations of Compact Groups 38
Chapter 2 Analogues of the Beltrami-Klein Model for Rank One Symmetric Spaces of Noncompact Type 43
The Real Hyperbolic Space SO0(n,1)/SO(n) 44
The Complex Hyperbolic Space SU(n,1)/S(U(n)xU(1)) 51
The Quaternionic Hyperbolic Space Sp(n,1)/Sp(n)xSp(1) 55
The Cayley Hyperbolic Plane F*4/Spin(9) 60
Chapter 3 Realizations of Rank One Symmetric Spaces of Compact Type 69
The Space SO(n+1)/SO(n) 69
The Real Projective Space SO(n+1)/O(n) 73
The Complex Projective Space SU(n+1)/S(U(n)xU(1)) 79
The Quaternionic Projective Space Sp(n+1)/Sp(n)xSp(1) 83
The Cayley Projective Plane F4/Spin(9) 87
Chapter 4 Realizations of the Irreducible Components of the Quasi-Regular Representation of Groups Transitive on Spheres. Invariant Subspaces 92
The Groups SO(n) and O(n) 93
The Group U(n) 98
The Group OC(n) 101
The Group Sp(n) 105
The Group OQ(n) 128
The Group OCa(2) 131
Chapter 5 Non-Euclidean Analogues of Plane Waves 142
The Case of HRn 143
The Case of HCn 148
The Case of HQn 150
The Case of HCa2 155
Comments, Further Results, and Open Problems 160
Part II Transformations with Generalized Transmutation Property Associated with Eigenfunctions Expansions 164
Chapter 6 Preliminaries 166
Holomorphic Functions 167
Distributions 176
Chapter 7 Some Special Functions 182
Cylindrical Functions 182
Jacobi Functions 185
Extension of the Jacobi Polynomials 193
Confluent Hypergeometric Functions 200
Chapter 8 Exponential Expansions 206
Main Classes of Distributions 207
Biorthogonal Systems. General Completeness Results 212
Expansions in Series of Exponentials 224
The Distribution zetaT. Solution of the Lyubich Problem 230
Chapter 9 Multidimensional Euclidean Case 235
Introductory Results 236
Spherical Functions and Their Generalizations 239
Hankel-Like Integral Transforms 247
Transmutation Operators Induced by the Converse Hankel Transform. Connection with the Dual Abel Transform 250
Bessel-Type Decompositions for Some Classes of Functions with Generalized Boundary Conditions 261
Chapter 10 The Case of Symmetric Spaces X=G/K of Noncompact Type 272
Generalities 272
Fourier Decompositions on G/K 277
Eisenstein-Harish-Chandra Integrals and Their Rank One Generalizations 281
The Helgason-Fourier Transform f(lambda,b) 293
Action of f(lambda,b) on the Space E'delta(X) 297
The Transmutation Mapping Adelta Related to the Inversion Formula for the delta-Spherical Transform 303
The Class E'(X) of Distributions with Radial Spherical Transform. Mean Value Characterization. Explicit Form for X=G/K (G Complex) 312
Some Rank One Results on the Mapping Adelta 316
Ideas and Methods of Sect. 9.5 Applied in Analogous Problems for G/K 329
Chapter 11 The Case of Compact Symmetric Spaces 338
Compact Symmetric Spaces of Rank One from the Point of View of Realizations 339
Continuous Family of Eigenfunctions of the Laplace-Beltrami Operator 345
Analytic Extension of the Discrete Fourier-Jacobi Transform 356
The Transmutation Operators Ak,m,j Associated with the Jacobi Polynomials Expansion 363
Analogues of Ak,m,j in Exterior of a Ball. The Zaraisky Theorem 368
Chapter 12 The Case of Phase Space 374
The Twisted Convolution of Distributions on Cn. Special Hermite Operator 375
Expansions over Bigraded Spherical Harmonics 377
Derivatives of Generalized Laguerre Functions 380
Analogues of the Spherical Transform 387
Transmutation Mappings Generated by the Laguerre Polynomials Expansion 393
Comments, Further Results, and Open Problems 398
Part III Mean Periodicity 403
Chapter 13 Mean Periodic Functions on Subsets of the Real Line 405
Main Classes of Mean Periodic Functions 406
Structure of Zero Sets 408
Nonharmonic Fourier Series 416
Local Analogues of the Schwartz Fundamental Principle 424
The Problem of Mean Periodic Continuation 427
One-Sided Liouville's Property 438
Chapter 14 Mean Periodic Functions on Multidimensional Domains 441
General Properties 442
Modern Versions of the John Theorem. Connections with the Spectrum. The Hemisphere Theorem 447
Multidimensional Analogues of the Distribution zetaT. Mean Periodic Functions with Support in Exterior of a Ball. Exactness of Uniqueness Theorems 453
Analogues of the Taylor and the Laurent Expansions for Mean Periodic Functions. Estimates of the Coefficients 458
Convergence Theorems. Extendability and Nonextendability Results 469
Problem on Admissible Rate of Decreasing. Reduction to the Helmholtz Equation 474
Hörmander-Type Approximation Theorems on Domains Without the Convexity Assumption 480
Chapter 15 Mean Periodic Functions on G/K 487
Preliminary Results 487
Uniqueness Problem. Features for Higher Ranks 492
Refinements for the Rank One Case 495
Counterexamples to the Uniqueness Problem 500
Generalized Spherical Functions Series 503
Structure Theorems and Their Applications 510
Sharp Growth Estimates. Comparing with Eigenfunctions of the Laplacian 515
Chapter 16 Mean Periodic Functions on Compact Symmetric Spaces of Rank One 523
Group and Infinitesimal Properties 524
Uniqueness Results 535
Series Development Theorems 542
Chapter 17 Mean Periodicity on Phase Space and the Heisenberg Group 545
Background Material 546
Phase Space Analogues of the Uniqueness Theorems 548
Characterizations of the Kernel of the Twisted Convolution Operator 554
Comments, Further Results, and Open Problems 558
Part IV Local Aspects of Spectral Analysis and the Exponential Representation Problem 569
Chapter 18 A New Look at the Schwartz Theory 572
Localization of the Schwartz Theorems. The Effect of the Size of the Domain 573
Pairwise Mean Periodic Functions 579
The Case of ``Small'' Intervals. Connections with Division-Type Problems for Entire Functions 587
The Deconvolution Problem. Explicit Reconstruction Formulae 590
Chapter 19 Recent Developments in the Spectral Analysis Problem for Higher Dimensions 594
Solution of the Berenstein-Gay Problem. Generalizations 595
Expansions Associated with Cylindrical Functions 601
More on the Berenstein-Gay Problem: The Case R=< r1 +r2
The Equivalence of the Local and the Global Pompeiu Properties 608
Chapter 20 E'(X) Spectral Analysis on Domains of Noncompact Symmetric Spaces of Arbitrary Rank 612
Symmetric Space Analogues of the Local Version of the Brown-Schreiber-Taylor Theorem 612
E'(X) Mean Periodic Functions with Respect to a Couple of Distributions 619
Explicit Representation Theorems 623
Zalcman-Type Two-Radii Problems on Domains of G/K (G Complex) 626
Chapter 21 Spherical Spectral Analysis on Subsets of Compact Symmetric Spaces 629
The Case of the Whole Space. The Contrast Between the Noncompact Type and the Compact Type 629
Freak-Like Theorems on Subsets of Compact Symmetric Spaces of Rank One 631
Comments, Further Results, and Open Problems 635
Bibliography 643
Index 657
Author Index 665