Random Fourier Series with Applications to Harmonic Analysis. (AM-101), Volume 101

Michael B. Marcus; Gilles Pisier

Published by Princeton University Press 1982

Random Fourier Series with Applications to Harmonic Analysis. (AM-101), Volume 101

Michael B. Marcus and Gilles Pisier

Volume 101 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9781400881536

Overview
Contents

About this book

In this book the authors give the first necessary and sufficient conditions for the uniform convergence a.s. of random Fourier series on locally compact Abelian groups and on compact non-Abelian groups. They also obtain many related results. For example, whenever a random Fourier series converges uniformly a.s. it also satisfies the central limit theorem. The methods developed are used to study some questions in harmonic analysis that are not intrinsically random. For example, a new characterization of Sidon sets is derived.

The major results depend heavily on the Dudley-Fernique necessary and sufficient condition for the continuity of stationary Gaussian processes and on recent work on sums of independent Banach space valued random variables. It is noteworthy that the proofs for the Abelian case immediately extend to the non-Abelian case once the proper definition of random Fourier series is made. In doing this the authors obtain new results on sums of independent random matrices with elements in a Banach space. The final chapter of the book suggests several directions for further research.

Topics

Frontmatter Publicly Available Download PDF	i
CONTENTS Publicly Available Download PDF	v
CHAPTER I: INTRODUCTION Requires Authentication Unlicensed Licensed Download PDF	1
CHAPTER II: PRELIMINARIES Requires Authentication Unlicensed Licensed Download PDF	16
CHAPTER III: RANDOM FOURIER SERIES ON LOCALLY COMPACT ABELIAN GROUPS Requires Authentication Unlicensed Licensed Download PDF	51
CHAPTER IV: THE CENTRAL LIMIT THEOREM AND RELATED QUESTIONS Requires Authentication Unlicensed Licensed Download PDF	65
CHAPTER V: RANDOM FOURIER SERIES ON COMPACT NON-ABELIAN GROUPS Requires Authentication Unlicensed Licensed Download PDF	74
CHAPTER VI: APPLICATIONS TO HARMONIC ANALYSIS Requires Authentication Unlicensed Licensed Download PDF	105
CHAPTER VII: ADDITIONAL RESULTS AND COMMENTS Requires Authentication Unlicensed Licensed Download PDF	122
REFERENCES Requires Authentication Unlicensed Licensed Download PDF	144
INDEX OF TERMINOLOGY Requires Authentication Unlicensed Licensed Download PDF	148
INDEX OF NOTATIONS Requires Authentication Unlicensed Licensed Download PDF	149
Backmatter Requires Authentication Unlicensed Licensed Download PDF	151