Approximation Theory: From Taylor Polynomials to Wavelets / Edition 1

by Ole Christensen, Khadija Laghrida Christensen

Overview

This book gives an elementary introduction to a classical area of mathematics - approximation theory - in a way that naturally leads to the modern field of wavelets. The main thread throughout the book is the idea of ap­ proximating "complicated expressions" with "simpler expressions," and how this plays a decisive role in many areas of modern mathematics and its applications. One of the main goals of the presentation is to make it clear to the reader that mathematics is a subject in a state of continuous evolution. This fact is usually difficult to explain to students at or near their second year of uni­ versity. Often, teachers do not have adequate elementary material to give to students as motivation and encouragement for their further studies. The present book will be of use in this context because the exposition demonstrates the dynamic nature of mathematics and how classical disciplines influence many areas of modern mathematics and applications. The book may lead readers toward more advanced literature, such as the other pub­ lications in the Applied and Numerical Harmonic Analysis series (ANHA), by introducing ideas presented in several of those books in an elementary context. The focus here is on ideas rather than on technical details, and the book is not primarily meant to be a textbook.

Product Details

ISBN-13:	9780817636005
Publisher:	Birkhäuser Boston
Publication date:	03/18/2004
Series:	Applied and Numerical Harmonic Analysis
Edition description:	2004
Pages:	156
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

About the Author

Ole Christensen is the author of An Introduction to Frames and Riesz Bases (0-8176-4295-1).

Table of Contents

1 Approximation with Polynomials.- 1.1 Approximation of a function on an interval.- 1.2 Weierstrass’ theorem.- 1.3 Taylor’s theorem.- 1.4 Exercises.- 2 Infinite Series.- 2.1 Infinite series of numbers.- 2.2 Estimating the sum of an infinite series.- 2.3 Geometric series.- 2.4 Power series.- 2.5 General infinite sums of functions.- 2.6 Uniform convergence.- 2.7 Signal transmission.- 2.8 Exercises.- 3 Fourier Analysis.- 3.1 Fourier series.- 3.2 Fourier’s theorem and approximation.- 3.3 Fourier series and signal analysis.- 3.4 Fourier series and Hilbert spaces.- 3.5 Fourier series in complex form.- 3.6 Parseval’s theorem.- 3.7 Regularity and decay of the Fourier coefficients.- 3.8 Best N-term approximation.- 3.9 The Fourier transform.- 3.10 Exercises.- 4 Wavelets and Applications.- 4.1 About wavelet systems.- 4.2 Wavelets and signal processing.- 4.3 Wavelets and fingerprints.- 4.4 Wavelet packets.- 4.5 Alternatives to wavelets: Gabor systems.- 4.6 Exercises.- 5 Wavelets and their Mathematical Properties.- 5.1 Wavelets and L2 (?).- 5.2 Multiresolution analysis.- 5.3 The role of the Fourier transform.- 5.4 The Haar wavelet.- 5.5 The role of compact support.- 5.6 Wavelets and singularities.- 5.7 Best N-term approximation.- 5.8 Frames.- 5.9 Gabor systems.- 5.10 Exercises.- Appendix A.- A.1 Definitions and notation.- A.2 Proof of Weierstrass’ theorem.- A.3 Proof of Taylor’s theorem.- A.4 Infinite series.- A.5 Proof of Theorem 3 7 2.- Appendix B.- B.1 Power series.- B.2 Fourier series for 2?-periodic functions.- List of Symbols.- References.

Editorial Reviews

"Approximation methods and wavelets have found many important applications in signal processing. This well-written textbook provides an elementary introduction to approximation theory. The authors concentrate on the presentation of main ideas and demonstrate the great influence of classical approximation theory on modern applied mathematics. This book, with many illustrative examples, is easy to read…[and] will be very useful for anyone interested in approximation theory and wavelets." —Zentralblatt Math

From the Publisher