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

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, 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.