Introduction to Fourier Analysis and Wavelets (Graduate Studies in Mathematics) [Hardcover]

Mark A. Pinsky (Author)

Book Description

ISBN-10: 082184797X | ISBN-13: 978-0821847978 | Publication Date: February 18, 2009

This book provides a concrete introduction to a number of topics in harmonic analysis, accessible at the early graduate level or, in some cases, at an upper undergraduate level. Necessary prerequisites to using the text are rudiments of the Lebesgue measure and integration on the real line. It begins with a thorough treatment of Fourier series on the circle and their applications to approximation theory, probability, and plane geometry (the isoperimetric theorem). Frequently, more than one proof is offered for a given theorem to illustrate the multiplicity of approaches. The second chapter treats the Fourier transform on Euclidean spaces, especially the author's results in the three-dimensional piecewise smooth case, which is distinct from the classical Gibbs-Wilbraham phenomenon of one-dimensional Fourier analysis. The Poisson summation formula treated in Chapter 3 provides an elegant connection between Fourier series on the circle and Fourier transforms on the real line, culminating in Landau's asymptotic formulas for lattice points on a large sphere. Much of modern harmonic analysis is concerned with the behavior of various linear operators on the Lebesgue spaces $L^p(\mathbb{R}^n)$. Chapter 4 gives a gentle introduction to these results, using the Riesz-Thorin theorem and the Marcinkiewicz interpolation formula. One of the long-time users of Fourier analysis is probability theory. In Chapter 5 the central limit theorem, iterated log theorem, and Berry-Esseen theorems are developed using the suitable Fourier-analytic tools. The final chapter furnishes a gentle introduction to wavelet theory, depending only on the $L_2$ theory of the Fourier transform (the Plancherel theorem). The basic notions of scale and location parameters demonstrate the flexibility of the wavelet approach to harmonic analysis. The text contains numerous examples and more than 200 exercises, each located in close proximity to the related theoretical material. Originally published by Brooks Cole/Cengage Learning as ISBN: 978-0-534-37660-4.

Product Details

• Hardcover: 376 pages
• Publisher: American Mathematical Society (February 18, 2009)
• Language: English
• ISBN-10: 082184797X
• ISBN-13: 978-0821847978
• Product Dimensions: 9.3 x 6.4 x 0.8 inches
• Shipping Weight: 1.4 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #1,204,255 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

5 of 6 people found the following review helpful:

5.0 out of 5 stars Fourier Analysis and Wavelets for Everyone, May 28, 2008

By 

R. S. Wenocur - See all my reviews
(REAL NAME)   

This review is from: Introduction to Fourier Analysis and Wavelets (Brooks/Cole Series in Advanced Mathematics) (Hardcover)

As a probabilist and statistician, with a Ph.D., having worked at universities, as a consultant, and in industry for approximately forty years, I had previously employed Fourier Analysis only as a tool, not having studied the subject as a discipline unto itself. Dr. Pinsky's book has allowed me to learn the subject more deeply, and from a different, exciting viewpoint. In addition, I needed a resource that would permit me to learn about wavelets. How wonderful to find a book that includes both topics! Moreover, this book is a pleasure to read, with pencil and paper, to work through the ideas. It is extraordinarily well-written, which is not surprising, given the clarity and excellence of Dr. Mark A. Pinsky's other works. Pinsky's grammar is excellent, which is extremely refreshing. Many modern authors cause me to believe that I should have a red pen to correct grammatical errors while reading their works. This book is much more than just a textbook; it is beautiful mathematically and beautifully written.

8 of 12 people found the following review helpful:

5.0 out of 5 stars For the Students!, July 22, 2002

By 

Palle E T Jorgensen "Palle Jorgensen" (Iowa City, Iowa United States) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: Introduction to Fourier Analysis and Wavelets (Brooks/Cole Series in Advanced Mathematics) (Hardcover)

Courses in harmonic analysis have a central place in the course offerings of every math department, be it pure or applied;-- and the subject is as important as ever! Yet it has not always been easy for an instructor to find a book that is right for the students. Some books might be too skimpy on proofs, or not deep enough.-- Or the applications may somehow be artificial, or contrived. Afterall, we teach the material to engineers!-- It is a relief to find, in Pinsky's lovely new book, a balanced approach to the subject. The motivation and the history receive a beautiful presentation, as do the technical points and proofs. And the historical comments- sprinkled throughout the book- bring the subject to life. At the same time, the book is forward looking, and it has been tested in courses. Great exercises! The structure of the exposition is friendly, and gently leads the reader toward the exciting new wavelet material in the last hundred or so pages of the book. The student thereby gets a sense of how the central questions in wavelet theory have their root in the more classical ideas of harmonic analysis.

1 of 6 people found the following review helpful:

5.0 out of 5 stars Excellent textbook and reference, which is readable!, June 17, 2008

By 

Mathematics Student "Diana" - See all my reviews

This review is from: Introduction to Fourier Analysis and Wavelets (Brooks/Cole Series in Advanced Mathematics) (Hardcover)

I needed to learn Fourier Analysis and Wavelets, and this book is excellent as a textbook and as a reference. It is also quite readable. We need more mathematics books like this one.