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Lectures on Hermite and Laguerre Expansions. (MN-42), Volume 42 : Mathematical Notes - Sundaram Thangavelu

Lectures on Hermite and Laguerre Expansions. (MN-42), Volume 42

Mathematical Notes


Published: 1st April 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.

Hermite, Special Hermite and Laguerre Functions
Hermite and Laguerre Functionsp. 1
Heisenberg Group and Weyl Transformp. 9
Special Hermite Functionsp. 14
Hermite and Laguerre Functions on the Heisenberg Groupp. 22
Asymptotic Properties of Hermite and Laguerre Functionsp. 26
Special Hermite Expansions
Special Hermite Expansionsp. 29
Oscillatory Singular Integrals and Riesz Transformsp. 32
Littlewood-Paley-Stein Theoryp. 39
Multipliers for Special Hermite Expansionsp. 45
Riesz Means for Special Hermite Expansionsp. 51
Proof of the Restriction Theoremp. 59
Multiple Hermite Expansions
Riesz Means and the Critical Indexp. 65
An L[superscript 2] for the Riesz Kernelp. 69
Almost Everywhere and Mean Summabilityp. 77
Hermite Expansions for Radial Functionsp. 81
Multipliers for Hermite Expansions
Littlewood-Paley-Stein Theory for the Hermite Semigroupp. 85
Marcinkiewicz Multiplier Theoremp. 90
Conjugate Poisson Integrals and Riesz Transformsp. 94
Wave Equation for the Hermite Operatorp. 105
Hermite Expansions on R
Cesaro Means and the Critical Indexp. 111
An Expression for the Cesaro Kernel and the Basic Estimatesp. 114
Estimating the Oscillatory Integralsp. 119
Almost Everywhere and Mean Summability Resultsp. 131
Laguerre Expansions
The Laguerre Convolutionp. 137
Laguerre Expansions of Convolution Typep. 143
Standard Laguerre Expansionsp. 150
Laguerre Expansions of Hermite Typep. 156
The Transplantation Theorems
The Transplantation Operatorsp. 167
Further Reduction in the Proof of the Propositionsp. 171
Some Preliminary Lemmasp. 176
Estimating [actual symbol not reproducible]p. 181
Estimating [actual symbol not reproducible]p. 186
Bibliographyp. 193
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691000480
ISBN-10: 0691000484
Series: Mathematical Notes
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 214
Published: 1st April 1993
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 22.9 x 15.2  x 1.5
Weight (kg): 0.31