Rational trigonometric approximations using Fourier series partial sums

A class of approximations {SN,M} to a periodic functionf which uses the ideas of Padé, or rational function, approximations based on the Fourier series representation off, rather than on the Taylor series representation off, is introduced and studied. Each approximationSN,M is the quotient of a trigonometric polynomial of degreeN and a trigonometric polynomial of degreeM. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients ofSN,M agree with those off. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients off. It is proven that these “Fourier-Padé” approximations converge point-wise to (f(x+) +f(x−))/2 more rapidly (in some cases by a factor of 1/k2M) than the Fourier series partial sums on which they are based. The approximations are illustrated by several examples and an application to the solution of an initial, boundary value problem for the simple heat equation is presented.