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Fourier Analysis: An Introduction (Princeton Lectures in Analysis) [Hardcover]

by Elias M. Stein, Rami Shakarchi
4.4 out of 5 stars  See all reviews (8 customer reviews)

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Book Description

April 6, 2003 069111384X 978-0691113845

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.

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Fourier Analysis: An Introduction (Princeton Lectures in Analysis) + Complex Analysis (Princeton Lectures in Analysis, No. 2) + Real Analysis: Measure Theory, Integration, and Hilbert Spaces (Princeton Lectures in Analysis) (Bk. 3)
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Editorial Reviews

About the Author

Elias M. Stein is Professor of Mathematics at Princeton University. Rami Shakarchi received his Ph.D. in Mathematics from Princeton University in 2002.

Product Details

  • Series: Princeton Lectures in Analysis (Book 1)
  • Hardcover: 320 pages
  • Publisher: Princeton University Press (April 6, 2003)
  • Language: English
  • ISBN-10: 069111384X
  • ISBN-13: 978-0691113845
  • Product Dimensions: 9.6 x 6.3 x 1 inches
  • Shipping Weight: 1.3 pounds (View shipping rates and policies)
  • Average Customer Review: 4.4 out of 5 stars  See all reviews (8 customer reviews)
  • Amazon Best Sellers Rank: #104,388 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
25 of 27 people found the following review helpful
4.0 out of 5 stars Challenging July 25, 2003
I have just finished a class with the book as its main textbook. The book is well written, but you honestly have to work through each page with pen and paper in hand filling in the omitted steps. Nothing is spoon-fed to you. The exercises are very challenging while the problems develop small theories. If you work through the pain and sweat through the exercises, you will at the end of the book greatly improve your skills and intuition.
The author Stein is a leader in his field and has provided plenty of depth and breadth. This also means that he is on a different level and an argument that he calls "simple" has quite often taken me two pages to justify. However, if you put in the effort it will pay off tenfold.
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21 of 23 people found the following review helpful
5.0 out of 5 stars Excellent, if you've got some experience in analysis December 18, 2004
By ch0ni
I used this book for an undergraduate-level course in Fourier analysis. It is an excellent text, although I would recommend the prospective learner to take a basic course in real analysis first (or perhaps concurrently, if the learner dares!). With my experience in analysis, it proved very readable. In fact, it strengthened my understanding of (and even interest in!) analysis, as it provides a fruitful application of the subject--one gets to see various important analysis ideas and techniques used in context. One could almost say that the text is an excellent complement to real analysis to help the ideas jell. On the other hand, perhaps it is theoretically possible to use this book as a springboard into learning analysis. The proofs do gloss over some details, which as the previous reviewer noted, can make things tough going at times... I actually found this useful (again, perhaps because of analysis experience), as it omits just enough detail to stay focused on the subject at hand (being too pedantic is likely to make those of shorter attention spans, such as myself, want to wander away), and yet supplies enough detail to remind the reader of the underlying theory, and that all this stuff is mathematically rigorously justified.

The course I took was actually a brand-new course created at the undergraduate level, and was structured around the book, which had also just come out at the time. I can say with confidence that the course was a success, which is pretty unusual for something hot off the press (true, the book itself was based on lectures, but every university has its quirks...).
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15 of 17 people found the following review helpful
This is a somewhat biased review because sometimes I find myself searching for a good reference that treats a subject matter that is well-known in an easy, direct and accessible way. When I find such a book I end up relieved. This is what happened with the book by Stein and Shakarchi titled "Fourier Analysis".

In my case the search was for easy and accessible treatement of the theory of distributions in general and its applications to the wave equation in particular.

There are a number of references that treat this subject matter but all the ones I know of do this from a more advanced point of view. Stein and Shakarchi's book stems from an undergraduate lecture sequence thought at Princeton and the level of the text is indeed appropriate for the bright undergraduate who may or may not major in mathematics later on.

This is unlike PDE books by Taylor, or lecture notes by Melrose, or even the tiny booklet by Friedlander and Joshi that introduce distributions and their application to PDEs (like the wave equation) and certainly unlike Hörmanders comprehensive 4-volume treatment of the whole subject matter. All these references shoot significantly higher in terms of technical sophistication and I'd certainly not recommend them to typical engineering students for self-study. As possible exception I might mention Shubin's PDE books and encyclopedia contributions but they are more terse than the book under review and give less ground to more introductory matters.

Not so the book under review.
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3 of 3 people found the following review helpful
5.0 out of 5 stars very good but it is not an introduction July 15, 2007
By areader
Format:Hardcover|Verified Purchase
This is a very nice book in Fourier analysis with strong applications or examples in elementary partial differential equations. It is the first book of the three volumes set in the Princeton Lectures in Analysis. However, it is not an introductory text and some background in elementary analysis is required to fully appreciate its content.
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