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:

• The emphasis is on topics close to numerical algorithms.
• Everything is illustrated with Chebfun.
• Each chapter is a publishable Matlab M-file, available online.
• There is bias towards theorems and methods for analytic functions, which appear so often in applications, rather than on functions at the edge of discontinuity with their seductive theoretical challenges.
• Original sources are cited rather than textbooks, and each item in the bibliography is accompanied by an editorial comment.

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