This is a textbook on classical polynomial and rational approximation theory for the twenty-first century. It is aimed at advanced undergraduates and graduate students across all of applied mathematics. The following are the distinctive features of the book:
Keywords: approximation theory, numerical analysis, quadrature, spectral methods
Nick Trefethen is Professor of Numerical Analysis at the University of Oxford and a Fellow of the Royal Society. During 2011-2012 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. Carathéodory–Fejér 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; Appendix: Six Myths of Polynomial Interpolation and Quadrature; References; Index
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