This concisely written book gives an elementary introduction to a classical area of mathematicsapproximation theoryin a way that naturally leads to the modern field of wavelets. The exposition, driven by ideas rather than technical details and proofs, demonstrates the dynamic nature of mathematics and the influence of classical disciplines on many areas of modern mathematics and applications. Included are classical, illustrative examples and constructions, exercises, and a discussion of the role of wavelets to areas such as digital signal processing and data compression.

One of the few books to describe wavelets in words rather than mathematical symbols, the work will be an excellent textbook or self-study reference for advanced undergraduate/beginning graduate students and instructors in pure and applied mathematics, mathematical physics, and engineering. Readers will find motivation and background material pointing toward advanced literature and research topics in pure and applied harmonic analysis and related areas.

Concisely written, user-friendly book Demonstrates the dynamic nature of mathematics and the influence of classical disciplines on many areas of modern mathematics and applications Includes classical, illustrative examples and constructions, exercises, and a discussion of the role of wavelets to areas such as digital signal processing and data compression Includes supplementary material: sn.pub/extras

Autorentext

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

Klappentext

This concisely written 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 exposition, driven by ideas rather than technical details and proofs, demonstrates the dynamic nature of mathematics and the influence of classical disciplines on many areas of modern mathematics and applications.

Key features and topics:

• Description of wavelets in words rather than mathematical symbols

• Elementary introduction to approximation using polynomials (Weierstrass' and Taylor's theorems)

• Introduction to infinite series, with emphasis on approximation-theoretic aspects

• Introduction to Fourier analysis

• Numerous classical, illustrative examples and constructions

• Discussion of the role of wavelets in digital signal processing and data compression, such as the FBI's use of wavelets to store fingerprints

• Minimal prerequisites: elementary calculus

• Exercises that may be used in undergraduate and graduate courses on infinite series and Fourier series

  Approximation Theory: From Taylor Polynomials to Wavelets will be an excellent textbook or self-study reference for students and instructors in pure and applied mathematics, mathematical physics, and engineering. Readers will find motivation and background material pointing toward advanced literature and research topics in pure and applied harmonic analysis and related areas.

  Zusammenfassung
  This book gives an elementary introduction to a classical area of mathemat­ ics - 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 demon­ strates 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.

  Inhalt
  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.