Approximation Theory

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

"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

• 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.