Lectures on Hermite and Laguerre Expansions

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ISBN: 9780691000480 | 0691000484
Cover: Paperback
Copyright: 4/19/1993

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The interplay between analysis on Lie groups and the theory of special functions is well known. This monograph deals with the case of the Heisenberg group and the related expansions in terms of Hermite, special Hermite, and Laguerre functions. The main thrust of the book is to develop a concrete Littlewood-Paley-Stein theory for these expansions and use the theory to prove multiplier theorems. The questions of almost-everywhere and mean convergence of Bochner-Riesz means are also treated. Most of the results in this monograph appear for the first time in book form.

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Preface
Acknowledgements
Hermite, Special Hermite and Laguerre Functions
Hermite and Laguerre Functions	p. 1
Heisenberg Group and Weyl Transform	p. 9
Special Hermite Functions	p. 14
Hermite and Laguerre Functions on the Heisenberg Group	p. 22
Asymptotic Properties of Hermite and Laguerre Functions	p. 26
Special Hermite Expansions
Special Hermite Expansions	p. 29
Oscillatory Singular Integrals and Riesz Transforms	p. 32
Littlewood-Paley-Stein Theory	p. 39
Multipliers for Special Hermite Expansions	p. 45
Riesz Means for Special Hermite Expansions	p. 51
Proof of the Restriction Theorem	p. 59
Multiple Hermite Expansions
Riesz Means and the Critical Index	p. 65
An L[superscript 2] for the Riesz Kernel	p. 69
Almost Everywhere and Mean Summability	p. 77
Hermite Expansions for Radial Functions	p. 81
Multipliers for Hermite Expansions
Littlewood-Paley-Stein Theory for the Hermite Semigroup	p. 85
Marcinkiewicz Multiplier Theorem	p. 90
Conjugate Poisson Integrals and Riesz Transforms	p. 94
Wave Equation for the Hermite Operator	p. 105
Hermite Expansions on R
Cesaro Means and the Critical Index	p. 111
An Expression for the Cesaro Kernel and the Basic Estimates	p. 114
Estimating the Oscillatory Integrals	p. 119
Almost Everywhere and Mean Summability Results	p. 131
Laguerre Expansions
The Laguerre Convolution	p. 137
Laguerre Expansions of Convolution Type	p. 143
Standard Laguerre Expansions	p. 150
Laguerre Expansions of Hermite Type	p. 156
The Transplantation Theorems
The Transplantation Operators	p. 167
Further Reduction in the Proof of the Propositions	p. 171
Some Preliminary Lemmas	p. 176
Estimating [actual symbol not reproducible]	p. 181
Estimating [actual symbol not reproducible]	p. 186
Bibliography	p. 193
Table of Contents provided by Blackwell. All Rights Reserved.

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Lectures on Hermite and Laguerre Expansions

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