Abstract
A family of simple, periodic basis functions with “built-in” discontinuities are introduced, and their properties are analyzed and discussed. Some of their potential usefulness is illustrated in conjunction with the Fourier series representation of functions with discontinuities. In particular, it is demonstrated how they can be used to construct a sequence of approximations which converges exponentially in the maximum norm to a piece-wise smooth function. The theory is illustrated with several examples and the results are discussed in the context of other sequences of functions which can be used to approximate discontinuous functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Bary, N. K. (1964). A treatise on Trigonometric Series. MacMillan, New York.
Geer, J. (1995). Rational trigonometric approximations using Fourier series partial sums. ICASE Rep. No. 93-68 and J. Sci. Comp. 10, 325–356.
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. ICASE Rep. No. 92-4 and J. Comp. and Appl. Math. 43, 81–98.
Gottlieb, D., and Shu, C. (1994). Resolution properties of the Fourier method for discontinuous waves. ICASE Rep. No. 92-27 and Comp. Meth. Appl. Mech. Eng. 116, 27–37.
Gottlieb, D., and Shu, C. (1996a). On the Gibbs phenomena III: recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function. ICASE Rep. No. 93-82 and SIAM J. Num. Anal. 33, 280–290.
Gottlieb, D., and Shu, C. (1995). On the Gibbs phenomena IV: Recovering exponential accuracy in a sub-interval from a Gegenbauer partial sum of a piecewise analytic function. ICASE Rep. No. 94-33 and Math. of Comp. 64, 1081–1096.
Gottlieb, D., and Shu, C. (1996b). On the Gibbs phenomena V: Recovering exponential accuracy from collocation point values of a piecewise analytic function. ICASE Rep. No. 94-61 and Numerische Mathematic (to appear).
John, F. (1982). Partial Differential Equations, Springer-Verlag, New York.
Kreyszig, E. (1979). Advanced Engineering Mathematics (fourth edition). Wiley, New York.
Lanczos, C. (1966). Discourse on Fourier Series, Hafner Publishing, New York.
Thompson, W. J. (1992). Fourier series and the Gibbs phenomena. Am. J. Phys. 60(5), 425–429.
Tolstov, G. P. (1962). Fourier Series (translated from the Russian by R. A. Silverman), Prentice-Hall, New Jersey.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Geer, J., Banerjee, N.S. Exponentially Accurate Approximations to Piece-Wise Smooth Periodic Functions. Journal of Scientific Computing 12, 253–287 (1997). https://doi.org/10.1023/A:1025649427614
Issue Date:
DOI: https://doi.org/10.1023/A:1025649427614