This book presents a twenty-first century approach to classical polynomial and rational approximation theory. The reader will find a strikingly original treatment of the subject, completely unlike any of the existing literature on approximation theory, with a rich set of both computational and theoretical exercises for the classroom. There are many original features that set this book apart: the emphasis is on topics close to numerical algorithms; every idea is illustrated with Chebfun examples; each chapter has an accompanying Matlab file for the reader to download; the text focuses on theorems and methods for analytic functions; original sources are cited rather than textbooks, and each item in the bibliography is accompanied by an editorial comment. This textbook is ideal for advanced undergraduates and graduate students across all of applied mathematics.
Nick Trefethen is Professor of Numerical Analysis at the University of Oxford and a Fellow of the Royal Society. During 2011-12 he served as President of SIAM.
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. Caratheodory - 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. Pade approximation; 28. Analytic continuation and convergence acceleration; Appendix. Six myths of polynomial interpolation and quadrature; References; Index.
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