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MiS Preprint
75/2004
Approximation of $1/x$ by Exponential Sums in $[1,\infty)$
Dietrich Braess and Wolfgang Hackbusch
Abstract
Approximations of $1/x$ by sums of exponentials are well studied for finite intervals. Here the error decreases like $\mathcal{O}(\exp(-ck))$ with the order $k$ of the exponential sum. In this paper we investigate approximations of $1/x$ on the interval $[1,\infty)$. We prove estimates of the error by $\mathcal{O}(\exp(-c\sqrt{k}))$ and confirm this asymptotic estimate by numerical results. Numerical results lead to the conjecture that the constant in the exponent equals $c=\pi\sqrt{2}.$
