Graduate studies in mathematics· 2012
Higher Order Fourier Analysis
- 99citations
- 2012year
Short summary
This book introduces higher order Fourier analysis, a recent field extending traditional Fourier analysis with non-linear phase functions to study functions, particularly for problems involving arithmetic progressions.
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Key points
- Introduces higher order Fourier analysis, extending traditional methods with non-linear phase functions.
- Builds on Gowers' work for Szemeredi's theorem and connects to Weyl's and Furstenberg's theories.
- Covers applications to counting patterns in primes and serves as a graduate-level introduction.
- Includes numerous exercises for readers to test their understanding.
AI-generated from the title and abstract; the full text is not read.
Abstract
Traditional Fourier analysis, which has been remarkably effective in many contexts, uses linear phase functions to study functions. Some questions, such as problems involving arithmetic progressions, naturally lead to the use of quadratic or higher order phases. Higher order Fourier analysis is a subject that has become very active only recently. Gowers, in groundbreaking work, developed many of the basic concepts of this theory in order to give a new, quantitative proof of Szemeredi's theorem on arithmetic progressions. However, there are also precursors to this theory in Weyl's classical theory of equidistribution, as well as in Furstenberg's structural theory of dynamical systems. This book, which is the first monograph in this area, aims to cover all of these topics in a unified manner, as well as to survey some of the most recent developments, such as the application of the theory to count linear patterns in primes. The book serves as an introduction to the field, giving the beginning graduate student in the subject a high-level overview of the field. The text focuses on the simplest illustrative examples of key results, serving as a companion to the existing literature on the subject. There are numerous exercises with which to test one's knowledge.
The authors' abstract, as published at the source. Graduate studies in mathematics, 2012 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics