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Exponential Sum Approximations for t−β

In: Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Author

Listed:
  • William McLean

    (The University of New South Wales, School of Mathematics and Statistics)

Abstract

Given β > 0 and δ > 0, the function t −β may be approximated for t in a compact interval [δ, T] by a sum of terms of the form we−at, with parameters w > 0 and a > 0. One such an approximation, studied by Beylkin and Monzón (Appl. Comput. Harmon. Anal. 28:131–149, 2010), is obtained by applying the trapezoidal rule to an integral representation of t −β, after which Prony’s method is applied to reduce the number of terms in the sum with essentially no loss of accuracy. We review this method, and then describe a similar approach based on an alternative integral representation. The main difference is that the new approach achieves much better results before the application of Prony’s method; after applying Prony’s method the performance of both is much the same.

Suggested Citation

  • William McLean, 2018. "Exponential Sum Approximations for t−β," Springer Books, in: Josef Dick & Frances Y. Kuo & Henryk Woźniakowski (ed.), Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, pages 911-930, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-72456-0_40
    DOI: 10.1007/978-3-319-72456-0_40
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