Spectral methods and quadratures

Classical interpolation quadratures and, in particular, Gaussian quadratures are considered in the context of spectral methods, i.e., methods for solving boundary value problems for linear ODE by expanding them into series over orthogonal (and not only) polynomials. Fourier transforms are shown to play a key role here and allow calculating the required quadratures quite easily. Explicit formulas are given for some quadratures, and their efficiency is compared for high-accuracy computation of integrals. A simple Maple procedure for the Clenshaw–Curtis quadrature is given, and its application to computing the integral yielding the function of the sum of divisors of a natural number is considered.