Exponential sum approximations for $t^{-β}$

William McLean

doi:10.1007/978-3-319-72456-0_40

paper

Exponential sum approximations for $t^{- β}$

arXiv:1606.00123 · doi:10.1007/978-3-319-72456-0_40

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 $w e^{- a t}$ , with parameters $w > 0$ and $a > 0$ . One such an approximation, studied by Beylkin and Monzón, 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.

18 pages, 5 figures. I have completely rewritten this paper because after uploading the previous version I realised that there is a much better approach. Note the change to the title. Have included minor corrections following review

Exponential sum approximations for t−β

Abstract

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Exponential sum approximations for $t^{- β}$