Abstract
A class of approximations {S N,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 approximationS N,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 ofS N,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/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.
Similar content being viewed by others
References
Baker, G. A. (1975).Essentials of Padé Approximants, Academic Press, New York.
Bary, N. K. (1964).A Treatise on Trigonometric Series, MacMillan, New York.
Cheney, E. W. (1966).Introduction to Approximation Theory, McGraw-Hill, New York.
Clenshaw, C. W., and Lord, K. (1974). Rational approximations from Chebyshev series, InStudies in Numerical Analysis, Academic Press, New York, pp. 95–113.
Fike, C. T. (1968).Computer Evaluation of Mathematical Functions, Prentice-Hall, New Jersey.
Gottlieb, D., and Shu, C. (1993). Resolution properties of the Fourier method for discontinuous waves,Comp. Meth. Appl. Mech. Eng. (to appear).
Gottlieb, D., Shu, C., Solomonoff, A., and Vandeven, H. (1992). On the Gibbs phenomena I: recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function,J. Comp. and Appl. Math. 43, 81–98.
Gröbner, W., and Hofreiter, N. (1961).Integraltafel, Zweiter Teil, Bestimmte Integral, Springer-Verlag, Wien.
Maehly, H. J. (1960). Rational approximations for transcendental functions,Proc. Int. Conf. Information Processing, UNESCO, Butterworths, London.
Ralston, A. (1965).A First Course in Numerical Analysis, McGraw-Hill, New York.
Author information
Authors and Affiliations
Additional information
This research was supported by NASA contract NAS1-19480 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, Virginia 23665.
Rights and permissions
About this article
Cite this article
Geer, J.F. Rational trigonometric approximations using Fourier series partial sums. J Sci Comput 10, 325–356 (1995). https://doi.org/10.1007/BF02091779
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02091779