This book is intended as an introduction to harmonic analysis and generalized Gelfand pairs. Starting with the elementary theory of Fourier series and Fourier integrals, the author proceeds to abstract harmonic analysis on locally compact abelian groups and Gelfand pairs. Finally a more advanced theory of generalized Gelfand pairs is developed. This book is aimed at advanced undergraduates or beginning graduate students. The scope of the book is limited, with the aim of enabling students to reach a level suitable for starting PhD research. The main prerequisites for the book are elementary real, complex and functional analysis. In the later chapters, familiarity with some more advanced functional analysis is assumed, in particular with the spectral theory of (unbounded) self-adjoint operators on a Hilbert space. From the contents Fourier series Fourier integrals Locally compact groups Haar measures Harmonic analysis on locally compact abelian groups Theory and examples of Gelfand pairs Theory and examples of generalized Gelfand pairs
Author(s): Gerrit van Dijk
Series: De Gruyter Studies in Mathematics
Publisher: De Gruyter
Year: 2009
Language: English
Pages: 234
Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Preface ......Page 6
Contents ......Page 8
1.1 Definition and elementary properties ......Page 12
1.2 Convergence ......Page 13
1.3 Uniform convergence ......Page 14
1.5 Parseval?ˉs theorem ......Page 16
1.6 Generalization ......Page 17
2.1 The convolution product ......Page 18
2.2 Elementary properties of the Fourier integral ......Page 19
2.3 The inversion theorem ......Page 20
2.4 Plancherel?ˉs theorem ......Page 23
2.6 The Riemann¨CStieltjes integral and functions of bounded variation ......Page 24
2.7 Bochner?ˉs theorem ......Page 30
2.8 Extension to R^n ......Page 35
3.2 Topological spaces ......Page 36
3.3 Topological groups ......Page 38
3.4 Quotient spaces and quotient groups ......Page 40
3.5 Some useful facts ......Page 41
3.6 Functions on locally compact groups ......Page 42
4.1 Measures ......Page 43
4.2 Invariant measures ......Page 45
4.3 Weil?ˉs formula ......Page 50
4.5 Quasi-invariant measures on quotient spaces ......Page 52
4.6 The convolution product on G. Properties of L1.G/ ......Page 56
5.1 Positive-definite functions and unitary representations ......Page 60
5.2 Some functional analysis ......Page 66
5.3 Elementary positive-definite functions ......Page 71
5.4 Fourier transform, Riemann¨CLebesgue lemma and Bochner?ˉs theorem. ......Page 73
5.5 The inversion theorem ......Page 77
5.6 Plancherel?ˉs theorem ......Page 80
5.7 Pontryagin?ˉs duality theorem ......Page 81
5.8 Subgroups and quotient groups ......Page 84
5.9 Compact and discrete abelian groups ......Page 85
6.1 Gelfand pairs and spherical functions ......Page 86
6.2 Positive-definite spherical functions and unitary representations ......Page 90
6.3 Representations of class one ......Page 93
6.4 Harmonic analysis on Gelfand pairs ......Page 94
6.5 Compact Gelfand pairs ......Page 97
7.1 Euclidean motion groups ......Page 100
7.2 The sphere ......Page 106
7.3 Spherical harmonics ......Page 107
7.4 Spherical functions on spheres ......Page 116
7.5 Real hyperbolic spaces ......Page 117
8.1 C^1 vectors of a representation ......Page 142
8.2 Invariant Hilbert subspaces ......Page 146
8.3 Generalized Gelfand pairs ......Page 152
8.4 Invariant Hilbert subspaces of L2.G=H/ ......Page 156
9.1 Non-Euclidean motion groups ......Page 163
9.2 Pseudo-Riemannian real hyperbolic spaces ......Page 169
A.1 Special case of a theorem of Harish-Chandra ......Page 206
A.2 Results of M¨|th¨|e ......Page 208
A.3 Results of Tengstrand ......Page 211
A.4 Solutions in H of a singular second order differential equation ......Page 216
A.5 Expression of Mbf ./ in terms of Bessel functions ......Page 219
B.2 Analog of M¨|th¨|e?ˉs results ......Page 221
B.3 Tengstrand?ˉs results for X ......Page 223
B.4 Solutions in H of a singular second order differential equation ......Page 224
Bibliography ......Page 228
Index ......Page 232
