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Princeton Lectures in Analysis #1
Fourier Analysis: An Introduction
This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions.
The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression.
In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest.
The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.
The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression.
In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest.
The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.
328 pages, Hardcover
First published March 17, 2003
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Displaying 1 - 5 of 5 reviews
November 27, 2021
The authors of this book have provided some of the finest introductory books on mathematical analysis, and to whom I have become quite fond. Their style is elegant, well-organized, and distinguishably articulate. Problems are motivated in a manner that demonstrates their excellent scholarship and profound understanding of the subject, and their works would serve as excellent references for beginning and experienced mathematicians alike.
Nature teaches us math like no other, and perhaps that is manifest the most in Fourier analysis, a fascinating topic that I have been perpetually enchanted with throughout my undergraduate studies. All periodic functions can be decomposed into sines and cosines, permitting us to comprehend and thoroughly analyze them. This is an example of how mathematics granted us invaluable tools to read, understand, and control some aspects of nature. I have come to appreciate this in the field of electromagnetics, where we can see this beauty in its most breathtaking forms.
Nature teaches us math like no other, and perhaps that is manifest the most in Fourier analysis, a fascinating topic that I have been perpetually enchanted with throughout my undergraduate studies. All periodic functions can be decomposed into sines and cosines, permitting us to comprehend and thoroughly analyze them. This is an example of how mathematics granted us invaluable tools to read, understand, and control some aspects of nature. I have come to appreciate this in the field of electromagnetics, where we can see this beauty in its most breathtaking forms.
March 11, 2021
I quite enjoyed this mathematical treatment to Fourier analyses. The text is very clear and intuitive. It also contains several interesting applications of the Fourier analysis to other branches of mathematics.
April 20, 2016
This book is brimming with clarity and intuition. It develops basic Fourier analysis, and features *many* applications to other areas of mathematics. The proofs are elegant, the exercises terrific. It's one of the best books I have ever read.
October 11, 2023
Overall a great introduction to Fourier series, the transform, various types of convergence such as mean squared, Cesaro and Abel summability. What stands out are the applications to number theory e.g. finite Fourier analysis on Abelian groups, characters, primes...
As always, I'm not a huge fan of the usual mathematical style of starting with the definitions and theorems, which is backwards compared to the derivation or discovery process. More context and anecdotes would be helpful for mortals. I complemented this approach by also doing Stanford's oustanding class on the Fourier transform with Brad Osgood.
Overall a good book with some interesting exercises. I'll probably check out the next in the series.
As always, I'm not a huge fan of the usual mathematical style of starting with the definitions and theorems, which is backwards compared to the derivation or discovery process. More context and anecdotes would be helpful for mortals. I complemented this approach by also doing Stanford's oustanding class on the Fourier transform with Brad Osgood.
Overall a good book with some interesting exercises. I'll probably check out the next in the series.
March 20, 2024
I definitely understood this book yeah
Displaying 1 - 5 of 5 reviews