Pub. Date:
American Mathematical Society
Introduction to Fourier Analysis and Wavelets

Introduction to Fourier Analysis and Wavelets

by Mark A. Pinsky


Current price is , Original price is $73.0. You
Select a Purchase Option (New Edition)
  • purchase options
  • purchase options


Introduction to Fourier Analysis and Wavelets

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.

Product Details

ISBN-13: 9780821847978
Publisher: American Mathematical Society
Publication date: 02/18/2009
Series: Graduate Studies in Mathematics Series , #102
Edition description: New Edition
Pages: 376
Product dimensions: 6.40(w) x 9.30(h) x 0.80(d)

Table of Contents

1. Fourier Series on the Circle. Motivation and heuristics. Formulation of Fourier Series. Fourier Series in L2 . Norm Convergence and Summability. Improved Trigonometric Approximation. Divergence of Fourier Series.
Appendix: Complements on Laplace"s Method.
Appendix: Proof to the Uniform Boundedness theorem. Appendix-Higher-order Bessel functions. Appendix-Cantor"s uniqueness Ttheorem.
Appendix: Higher-Order Bessel Function.
Appendix: Cantor's Uniqueness Theorem.
2. Fourier Transforms on the Line and Space. Motivation and Heuristics. Basic Properties of the Fourier Transform. Fourier Inversion in One Dimension. L2 Theory in Rn. Spherical Fourier Inversion in Rn. Bessel Functions. The Method of Stationary Phase.
3. Fourier Analysis in Lp Spaces. Motivation and Heuristics. The M. Riesz-Thorin Interpolation Theorem. The Conjugate Function or Discrete Hilbert Transform. The Hilbert Transform on R. Hardy-Littlewood Maximal Function. The Marcinkiewicz Interpolation Theorem. Calderon-Zygmund Decomposition. A Class of Singular Integrals. Properties of Harmonic Functions.
4. Poisson Summation Formula and Multiple Fourier Series. Motivation and Heuristics. The Poisson Summation Formula in R1. Multiple Fourier Series. Poisson Summation Formula in Rd. Application to Lattice Points. Schrödinger Equation and Gauss Sums. Recurrence of Random walk.
5. Applications to Probability Theory. Motivation and Heuristics. Basic Definitions. Extension to Gap Series. Weak Convergence of Measures. Convolution Semigroups. The Berry-Esseen Theorem. The Law of the Iterated Logarithm.
6. Introduction to Wavelets. Motivation and Heuristics. WaveletTransform. Haar Wavelet Expansion. Multiresolution Analysis. Wavelets with Compact Support. Convergence Properties of Wavelet Expansions. Wavelet in Several Variables. References. Notations. Index.

Customer Reviews

Most Helpful Customer Reviews

See All Customer Reviews