On the Efficient Computation of High-Dimensional Integrals and the Approximation by Exponential Sums

Dietrich Braess and Wolfgang Hackbusch

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Submission date: 07. Jan. 2009 (revised version: November 2009)
Pages: 38
published as: On the efficient computation of high-dimensional integrals and the approximation by exponential sums.
In: Multiscale, nonlinear and adaptive approximation : dedicated to Wolfgang Dahmen on the occasion of his 60th birthday / R. A. DeVore ... (eds.)
Berlin [u.a.] : Springer, 2009. - P. 39 - 74 
DOI number (of the published article): 10.1007/978-3-642-03413-8_3
published as:
Clarke, Br.: The completion of the manifold of Riemannian metrics with respect to its L2 metric
   Dissertation, Universität Leipzig, 2009
Bibtex
MSC-Numbers: 11L07, 41A20
Keywords and phrases: exponential sums, rational functions, Chebyshev approximation, best approximation, completely monotone functions, Heron's algorithm, complete elliptic integrals, Landen transformation
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Abstract:
The approximation of the functions 1/x and by exponential sums enables us to evaluate some high-dimensional integrals by products of one-dimensional integrals. The degree of approximation can be estimated via the study of rational approximation of the square root function. The latter has interesting connections with the Babylonian method and Gauss' arithmetic-geometric process.