Approximation Theory and Approximation Practice

Contents: 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. Appendices. References. Index.

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.