Euler summation for fourier series and laplace transform inversion

Abstract

Euler summation is a convergence-acceleration technique which has proved very effective in Fourier-series methods for Laplace transform inversion. We present an analysis of the effect of Euler summation that explains its excellent performance. The central result is a bound on the truncation error when Euler summation is used. Our analysis supports some of the parameter-selection strategies proposed in Abate and Whitt's recent comprehensive treatment of Fourier series methods [2]. Specifically, we show that if our goal is to minimize the required number of terms of the Fourier series subject to the truncation error bound being no more than a specified target accuracy, then the “degree of averaging parameter” m should depend only on the desired accuracy, and not on properties of the function in question (so long as that function is sufficiently smooth). We also present a framework, here called “product smoothing, ” for constructing related summation methods with desirable properties