Spectral methods and quadratures

Classical interpolatory quadratures and, in particular, Gauss' quadratures are considered in the context of spectral methods, i.e., the methods of solution of boundary value problems for linear ODEs by expanding them in series of orthogonal (but not only) polynomials. We demonstrate that Fourier transforms play here a key role and allow to compute the needed quadratures in a simple way. We give explicit formulas for some quadratures and compare their effectiveness for high-precision computation of integrals. We present a simple Maple procedure for the Clenshaw-Curtis quadrature and consider its application to the computation of an integral giving the sum of divisors function.