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El. knyga: Analysis on h-Harmonics and Dunkl Transforms

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?This book provides an introduction to h-harmonics and Dunkl transforms. These are extensions of the ordinary spherical harmonics and Fourier transforms, in which the usual Lebesgue measure is replaced by a reflection-invariant weighted measure. The authors’ focus is on the analysis side of both h-harmonics and Dunkl transforms.Graduate students and researchers working in approximation theory, harmonic analysis, and functional analysis will benefit from this book.

Recenzijos

This well-written book gives a readable introduction to Dunkl harmonics and Dunkl transforms . the authors have collected a small compendium of results which will appeal to mathematicians interested in Dunkl analysis. The authors have done a commendable job in making this little book self-contained and quite readable. It will certainly serve as a starting point for graduate students and researchers interested in learning Dunkl harmonics and Dunkl transforms. (Sundaram Thangavelu, Mathematical Reviews, December, 2015)

Preface vii
1 Introduction: Spherical Harmonics and Fourier Transform
1(6)
1.1 Spherical harmonics
1(4)
1.2 Fourier transform
5(2)
2 Dunkl Operators Associated with Reflection Groups
7(8)
2.1 Weight functions invariant under a reflection group
7(3)
2.2 Dunkl operators
10(2)
2.3 Intertwining operator
12(2)
2.4 Notes and further results
14(1)
3 h-Harmonics and Analysis on the Sphere
15(20)
3.1 Dunkl h-harmonics
15(5)
3.2 Projection operator and intertwining operator
20(3)
3.3 Convolution operators and orthogonal expansions
23(4)
3.4 Maximal functions
27(4)
3.5 Convolution and maximal function
31(3)
3.6 Notes and further results
34(1)
4 Littlewood--Paley Theory and the Multiplier Theorem
35(16)
4.1 Vector-valued inequalities for self-adjoint operators
35(2)
4.2 The Littlewood--Paley--Stein function
37(2)
4.3 The Littlewood--Paley theory on the sphere
39(6)
4.3.1 A crucial lemma
40(2)
4.3.2 Proof of Theorem 4.3.3
42(3)
4.4 The Marcinkiewicz type multiplier theorem
45(2)
4.5 A Littlewood--Paley inequality
47(3)
4.6 Notes and further results
50(1)
5 Sharp Jackson and Sharp Marchaud Inequalities
51(14)
5.1 Introduction
51(1)
5.2 Moduli of smoothness and best approximation
52(2)
5.3 Weighted Sobolev spaces and K-functionals
54(2)
5.4 The sharp Marchaud inequality
56(3)
5.5 The sharp Jackson inequality
59(2)
5.6 Optimality of the power in the Marchaud inequality
61(1)
5.7 Notes and further results
62(3)
6 Dunkl Transform
65(30)
6.1 Dunkl transform: L2 theory
65(7)
6.2 Dunkl transform: L1 theory
72(4)
6.3 Generalized translation operator
76(6)
6.3.1 Translation operator on radial functions
77(3)
6.3.2 Translation operator for G = d2
80(2)
6.4 Generalized convolution and summability
82(5)
6.4.1 Convolution with radial functions
82(2)
6.4.2 Summability of the inverse Dunkl transform
84(2)
6.4.3 Convolution operator for Zd2
86(1)
6.5 Maximal function
87(7)
6.5.1 Boundedness of maximal function
87(3)
6.5.2 Convolution versus maximal function for Zd2
90(4)
6.6 Notes and further results
94(1)
7 Multiplier Theorems for the Dunkl Transform
95(16)
7.1 Introduction
95(1)
7.2 Proof of Theorem 7.1.1: part I
96(5)
7.3 Proof of Theorem 7.1.1: part II
101(4)
7.4 Proof of Theorem 7.1.1: part III
105(1)
7.5 Hormander's multiplier theorem and the Littlewood--Paley inequality
106(2)
7.6 Convergence of the Bochner--Riesz means
108(1)
7.7 Notes and further results
109(2)
Bibliography 111(6)
Index 117
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