Journal of Scientific Computing

, Volume 10, Issue 3, pp 325–356

Rational trigonometric approximations using Fourier series partial sums

  • James F. Geer

DOI: 10.1007/BF02091779

Cite this article as:
Geer, J.F. J Sci Comput (1995) 10: 325. doi:10.1007/BF02091779


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.

Key words

Fourier seriesrational approximationsGibbs phenomena

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • James F. Geer
    • 1
  1. 1.Department of Systems Science and Industrial Engineering, Watson School of Engineering and Applied ScienceState University of New YorkBinghamton