Overview
- Updated and self-contained exposition of all topics of Gaussian harmonic analysis that up to now are mostly scattered in research papers and sections of books
- Gaussian harmonic analysis may serve as a good introduction to other possible harmonic analysis done for orthogonal expansion
- A deep understanding of Gaussian harmonic analysis may help one gain insights into related problems in other non-euclidean settings
Part of the book series: Springer Monographs in Mathematics (SMM)
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About this book
Authored by a ranking authority in Gaussian harmonic analysis, this book embodies a state-of-the-art entrée at the intersection of two important fields of research: harmonic analysis and probability. The book is intended for a very diverse audience, from graduate students all the way to researchers working in a broad spectrum of areas in analysis. Written with the graduate student in mind, it is assumed that the reader has familiarity with the basics of real analysis as well as with classical harmonic analysis, including Calderón-Zygmund theory; also some knowledge of basic orthogonal polynomials theory would be convenient. The monograph develops the main topics of classical harmonic analysis (semigroups, covering lemmas, maximal functions, Littlewood-Paley functions, spectral multipliers, fractional integrals and fractional derivatives, singular integrals) with respect to the Gaussian measure. The text provide an updated exposition, as self-contained as possible, of all the topics in Gaussian harmonic analysis that up to now are mostly scattered in research papers and sections of books; also an exhaustive bibliography for further reading. Each chapter ends with a section of notes and further results where connections between Gaussian harmonic analysis and other connected fields, points of view and alternative techniques are given. Mathematicians and researchers in several areas will find the breadth and depth of the treatment of the subject highly useful.
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Keywords
- Gaussian measure
- Hermite polynomial expansions
- Ornstein-Uhlenbeck operator
- Ornstein-Uhlenbeck semigroup
- Poisson-Hermite semigroup
- covering lemmas for the Gaussian measure
- maximal functions with respect to the Gaussian measure
- Gaussian Littlewood-Paley functions
- Gaussian spectral multipliers
- Gaussian fractional integrals and fractional derivatives
- Gaussian singular integrals
Table of contents (10 chapters)
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Front Matter
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Back Matter
Reviews
“This book is the first comprehensive account of Gaussian harmonic analysis, and as such is a most welcome addition to the literature suitable for experts in the field and advanced graduate students who want to learn the subject.” (Jan van Neerven, Mathematical Reviews, September, 2020)
“This well-written and organized (mainly self-contained) book is a reader-friendly manual in the field of Gaussian harmonic analysis. It can be recommended for experts and for graduate, postgraduate and doctoral students.” (Michael Perelmuter, zbMATH 1421.42001, 2019)
“This well-written and organized (mainly self-contained) book is a reader-friendly manual in the field of Gaussian harmonic analysis. It can be recommended for experts and for graduate, postgraduate and doctoral students.” (Michael Perelmuter, zbMATH 1421.42001, 2019)
Authors and Affiliations
Bibliographic Information
Book Title: Gaussian Harmonic Analysis
Authors: Wilfredo Urbina-Romero
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-030-05597-4
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2019
Hardcover ISBN: 978-3-030-05596-7Published: 04 July 2019
eBook ISBN: 978-3-030-05597-4Published: 21 June 2019
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XIX, 477
Number of Illustrations: 4 b/w illustrations, 5 illustrations in colour
Topics: Abstract Harmonic Analysis