Next steps

The topics in this chapter come mainly under the heading of approximation theory, on which there are many good references. A thorough introduction to polynomial interpolation and approximation, emphasizing the complex plane and going well beyond the basics given here, is Trefethen (2013). A more thorough treatment of the least-squares case is given in Davis (1963).

A thorough comparison of Clenshaw–Curtis and Gauss–Legendre integration is given in Trefethen (2008).

The literature on the FFT is vast; a good place to start is with the brief and clear original paper by Cooley and Tukey Cooley & Tukey (1965). A historical perspective by Cooley on the acceptance and spread of the method can be found at the SIAM History Project at http://history.siam.org/cooley.htm (reprinted from Nash Nash (1990)). The FFT has a long and interesting history.

Doubly exponential integration, by contrast, is not often included in books. The original idea is presented in the readable paper Takahasi & Mori (1973), and the method is compared to Gaussian quadrature in Bailey et al. (2005), which is the source of some of the integration exercises in Improper integrals.