Random Fourier Series with Applications to Harmonic Analysis

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Princeton University Press, 1981 - Mathematics - 150 pages
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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.

 

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Contents

INTRODUCTION
3
PRELIMINARIES
16
2 A Jensen type inequality for the nondecreasing rearrangement of nonnegative stochastic processes
19
3 Continuity of Gaussian and subGaussian processes
24
4 Sums of Banach space valued random variables
40
RANDOM FOURIER SERIES ON LOCALLY COMPACT ABELIAN GROUPS
51
2 Random Fourier series on the real line
60
3 Random Fourier series on compact Abelian groups
63
3 Continuity of random Fourier series
93
APPLICATIONS TO HARMONIC ANALYSIS
105
2 Applications to Sidon sets
118
ADDITIONAL RESULTS AND COMMENTS
122
2 Almost sure almost periodicity
134
3 On left and right almost sure continuity
138
4 Generalizations
140
REFERENCES
144

THE CENTRAL LIMIT THEOREM AND RELATED QUESTIONS
65
RANDOM FOURIER SERIES ON COMPACT NONABELIAN GROUPS
74
2 Random series with coefficients in a Banach space
81

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About the author (1981)

fm.author_biographical_note1

Gilles Pisier is Emeritus Professor at the University of Paris VI, where he worked from 1981 to 2010. He is also a Distinguished Professor and holder of the Owen Chair in Mathematics at Texas A&M University. His international prizes include the Salem Prize in harmonic analysis (1979), the Ostrowski Prize (1997), and the Stefan Banach Medal (2001). He is a member of the Paris Academie des Sciences, a Foreign member of the Polish and Indian Academies of Science, and a Fellow of both the IMS and the AMS. He is also the author of several books, notably The Volume of Convex Bodies and Banach Space Geometry (Cambridge, 1989) and Introduction to Operator Space Theory (Cambridge, 2003).

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