Computation of Best L^∞ Exponential Sums for 1∕x by Remez’ Algorithm

Wolfgang Hackbusch

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Submission date: 18. Dec. 2017
Pages: 14
published in: Computing and visualization in science, 20 (2019) 1-2, p. 1-11 
DOI number (of the published article): 10.1007/s00791-018-00308-4
Bibtex
MSC-Numbers: 41A50, 65D15
Keywords and phrases: exponential sums, Remez algorithm, uniform best approximation
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Abstract:
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.