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Introduction to Fourier Analysis and Wavelets

Mark A. Pinsky Northwestern University, Evanston, IL
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Hardcover ISBN: 978-0-8218-4797-8
Product Code: GSM/102
List Price: $78.00 MAA Member Price:$70.20
AMS Member Price: $62.40 Electronic ISBN: 978-1-4704-1166-4 Product Code: GSM/102.E List Price:$73.00
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AMS Member Price: $93.60 Click above image for expanded view Introduction to Fourier Analysis and Wavelets Mark A. Pinsky Northwestern University, Evanston, IL Available Formats:  Hardcover ISBN: 978-0-8218-4797-8 Product Code: GSM/102  List Price:$78.00 MAA Member Price: $70.20 AMS Member Price:$62.40
 Electronic ISBN: 978-1-4704-1166-4 Product Code: GSM/102.E
 List Price: $73.00 MAA Member Price:$65.70 AMS Member Price: $58.40 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$117.00 MAA Member Price: $105.30 AMS Member Price:$93.60
• Book Details

Volume: 1022002; 376 pp
MSC: Primary 42;

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.

• Chapters
• Chapter 1. Fourier series on the circle
• Chapter 2. Fourier transforms on the line and space
• Chapter 3. Fourier analysis in $L^p$ spaces
• Chapter 4. Poisson simulation formula and multiple Fourier series
• Chapter 5. Applications to probability theory
• Chapter 6. Introduction to wavelets

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Volume: 1022002; 376 pp
MSC: Primary 42;

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.

• Chapter 3. Fourier analysis in $L^p$ spaces