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Approximation theory and approximation practice. (English) Zbl 1264.41001

Other Titles in Applied Mathematics 128. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-1-611972-39-9/pbk). 305 p. (2013).
From the introduction: “Approximation theory is an established field, and my aim is to teach you some of its most important ideas and results, centered on classical topics related to polynomials and rational functions. The style of this book, however, is quite different from what you will find elsewhere. Everything is illustrated computationally with the help of the Cebfun software package in Matlab, from Chebyshev interpolants to Lebesgue constants, from the Weierstrass approximation theorem to the Remez algorithm. Everything is practical and fast, so we will routinely compute polynomial interpolants or Gauss quadrature weights for tens of thousands points. In fact, each chapter of this book is a single Matlab M-file, and the book has been produced by executing these files with the Matlab ‘publish’ facility \(\ldots\) The book aims to be more readable than most, and the numerical experiments help achieve this. At the same time, theorems are stated and proofs are given, often rather tersely, without all the details spelled out. It is assumed that the reader is comfortable with rigorous mathematical arguments and familiar with ideas like continuous functions on compact set.”
The book is formed from twenty eight self-contained chapters: 1. Introduction; 2. Chebyshev points and interpolants; 3. Chebyshev polynomials and series; 4. Interpolants, projections, and aliasing; 5. Barycentric interpolation formula; 6. Weierstrass approximation theorem; 7. Convergence for differentiable functions; 8. Convergence for analytic functions; 9. Gibbs phenomenon; 10. Best approximation; 11. Hermite integral formula; 12. Potential theory and approximation; 13. Equispaced points, Runge phenomenon; 14. Discussion of high-order interpolation; 15. Lebesgue constants; 16. Best and near-best; 17. Orthogonal polynomials; 18. Polynomial roots and colleague matrices; 19. Clenshaw-Curtis and Gauss quadrature; 20. Carathéodory-Fejer approximation; 21. Spectral methods; 22. Linear approximation: beyond polynomials; 23. Nonlinear approximation: Why rational functions? 24. Rational best approximation; 25. Two famous problems; 26. Rational interpolation and linearized least-squares; 27. Padé approximation; 28. Analytic continuation and convergence acceleration.
“Virtually every chapter contains mathematical and scholarly novelties. Examples are the use of barycentric formulas beginning in Chapter 5, the tracing of barycentric formulas and the Hermite integral formula back to Jacobi in 1825 and Cauchy in 1826, Theorem 7.1 on the size of Chebyshev coefficients, the introduction to potential theory in Chapter 12, the discussion in Chapter 14 of prevailing misconceptions about interpolation, the presentation of colleague matrices for rootfinding in Chapter 18 with Jacobi matrices for quadrature as a special case in Chapter 19, Theorem 19.5 showing that Clenshaw-Curtis quadrature converges about as fast as Gauss quadrature, the first textbook presentation of Carathéodory-Fejer approximation in Chapter 20, the explanation in Chapter 22 of why polynomials are not optimal functions for linear approximation, the extensive discussion in Chapter 23 of the uses of rational approximations, and the SVD-based algorithms for robust rational interpolation and linearized least-squares fitting and Padé approximation in Chapters 26 and 27.”
Original sources are cited rather than textbooks, and each item in the bibliography is listed with an editorial comment. The book is primarily aimed at advanced undergraduates and graduate students across all of applied mathematics. It is a good source of information for all professionals interested in classical and modern methods of approximation theory.

MSC:

41-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to approximations and expansions
41-04 Software, source code, etc. for problems pertaining to approximations and expansions
41Axx Approximations and expansions
65Dxx Numerical approximation and computational geometry (primarily algorithms)

Software:

Chebfun; Matlab
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