Random Fourier Series with Applications to Harmonic Analysis

by Michael B. Marcus, Gilles Pisier

Overview

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.

Product Details

ISBN-13:	9780691082929
Publisher:	Princeton University Press
Publication date:	11/21/1981
Series:	Annals of Mathematics Studies , #101
Pages:	152
Product dimensions:	6.00(w) x 9.00(h) x (d)

Table of Contents

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