FourierAnalysis

This first volume of "The Princeton Lectures in Analysis" is a three-part introduction to Fourier analysis, intended for students with a basic understanding of mathematical analysis. The authors aim to introduce the key conceptual insights while providing the necessary technical underpinnings of rigorous analysis.

The first part begins with Fourier's simple conviction that arbitrary functions can be written as an infinite sum of basic trigonometric functions. This is explored in terms of convergence and summability of Fourier series, with applications such as the isoperimetric inequality and equidistribution.

The second part covers the Fourier transform and its applications to classical partial differential equations and the Radon transform. A clear introduction to the subject helps to avoid technical difficulties.

The book concludes with Fourier theory for finite abelian groups, including applications to prime numbers in arithmetic progression.

Throughout the volume, numerous examples and applications are provided to illustrate the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences.

Overall, students of mathematics, physics, engineering, and other sciences will find the theory and applications covered in this volume to be of real interest. This volume is the first of four planned volumes, which together aim to introduce the core areas of mathematical analysis while illustrating their organic unity. The subsequent volumes cover complex analysis, measure and integration theory, Hilbert spaces, functional analysis, distributions, and elements of probability theory.

作者简介

Stein在国际上享有盛誉，现任美国普林斯顿大学数学系教授。他是当代分析，特别是调和分析和分析领域领袖人物之一。古典调和分析最困难问题之一是推广到多维。他是多维欧氏调和分析的创造者之一，为此他发展了许多先进工具如奇异积分、Radon变换、极大函数等。他还发展了多个实变元的Hardy空间理论，推广了1971年F.John和L.Nirenberg的重要发现：即Hardy空间与BMO空间的对偶。在群上的调和分析方面也有贡献，例如同R.Kunze一起发现所谓Kunze-Stein现象。除此之外，他对多复变问题也做出了突出成绩。除了研究工作之外，他的许多书成为影响学科发展的重要参考文献。为此，他荣获1984年美国数学会在论述方面的Steele奖。由于他的成就，他在1974年被选为美国国家科学院院士，1982年被选为美国文理学院院士，1993年获得瑞士...