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MiS Preprint

Computation of Best $L^{\infty}$ Exponential Sums for $1/x$ by Remez' Algorithm

Wolfgang Hackbusch


The approximation of the function $1/x$ by exponential sums has several interesting applications. It is well known that best approximations with respect to the maximum norm exist. Moreover, the error estimates exhibit exponential decay as the number of terms increases. Here we focus on the computation of the best approximations. In principle, the problem can be solved by the Remez algorithm, however, because of the very sensitive behaviour of the problem the standard approach fails for a larger number of terms. The remedy described in the paper is the use of other independent variables of the exponential sum. We discuss the approximation error of the computed exponential sums up to 63 terms and hint to a webpage containing the corresponding coefficients.

Dec 18, 2017
Dec 19, 2017
MSC Codes:
41A50, 65D15
exponential sums, Remez algorithm, uniform best approximation

Related publications

2019 Journal Open Access
Wolfgang Hackbusch

Computation of best \(L^ \mathscr{\infty} \) exponential sums for \(1/x\) by Remez algorithm

In: Computing and visualization in science, 20 (2019) 1-2, pp. 1-11