Marcus / Pisier Random Fourier Series with Applications to Harmonic Analysis

1. Auflage 2016
ISBN: 978-1-4008-8153-6
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 101, 152 Seiten

Reihe: Annals of Mathematics Studies

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.

Autoren/Hrsg.

Marcus, Michael B.

Pisier, Gilles

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein

Inhaltsverzeichnis

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