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Numerical inversion for Laplace transforms of functions with discontinuities

Part of: Integral equations, integral transforms Special processes Probability theory and stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

T. Sakurai
T. Sakurai*
Affiliation:
University of Melbourne
*
∗ Postal address: ARC Special Research Centre for Ultra-Broadband Information Networks, Department of Electrical and Electronic Engineering, University of Melbourne, Melbourne, VIC 3010, Australia. Email address: t.sakurai@ee.mu.oz.au
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Abstract

We analyse the role of Euler summation in a numerical inversion algorithm for Laplace transforms due to Abate and Whitt called the EULER algorithm. Euler summation is shown to accelerate convergence of a slowly converging truncated Fourier series; an explicit bound for the approximation error is derived that generalizes a result given by O'Cinneide. An enhanced inversion algorithm called EULER-GPS is developed using a new variant of Euler summation. The algorithm EULER-GPS makes it possible to accurately invert transforms of functions with discontinuities at arbitrary locations. The effectiveness of the algorithm is verified through numerical experiments. Besides numerical transform inversion, the enhanced algorithm is applicable to a wide range of other problems where the goal is to recover point values of a piecewise-smooth function from the Fourier series.

Keywords

Numerical transform inversionEuler summationFourier seriesGibbs phenomenonconvergence accelerationLaplace transform

MSC classification

Primary: 60-04: Explicit machine computation and programs (not the theory of computation or programming)
Secondary: 60E10: Characteristic functions; other transforms 60K25: Queueing theory 65R10: Integral transforms
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 616 - 642
Copyright
Copyright © Applied Probability Trust 2004 

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