Rational Trigonometric Approximations Using Fourier Series Partial Sums

A class of approximation.s {,5'N,M } to a periodic function f which uses tile ideas of Pad6, or rational function, approximations based on the Fourier series representation of f, rather than oil the Taylor series representation of f, is introduced and studied. Each approximation ,q'X,M is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of ,gN,M agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these "Fourier-Pad6" approximations converge point-wise to (f(x +) + f(,r-))/2 more rapidly (in some cases by a factor of 1/k 2M) 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. 1This research was supported by the National Aeronautics and Space Administration under NASA (:ontract No. NAS1-19480 while the author was in residence at the Institute for (;omput.er Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681.