Approximation of \(1/\| x-y \|\) by exponentials for wavelet applications
type: Item
instance of: scholarly article
title: Approximation of \(1/\| x-y \|\) by exponentials for wavelet applications
zbMATH Open document ID: 1095.65107
author: Wolfgang Hackbusch
published in: Computing
publication date: 2006-03-02T00:00:00Z
review text: Integrals on high dimensional domains that occur in the solution of partial differential equations can often be transformed into products of lower dimensional integrals if \(1/\| x-y\| \) is approximated by exponentials. This leads to the approximation of the function \(1/\sqrt{x}\) by sums of exponentials. Here polynomial terms are added since this helps to control moments in connection with wavelets. Numerical results for the degree of approximation are provided for small and for large intervals in \(\mathbb R\). The improvement by the polynomial terms is discussed. The latter can be understood as exponential sums with fixed exponents. The tables are useful for a quick view of the error estimates [since the error estimates in the cited papers are less direct and require elliptic functions or the execution of Landen transformations].
Mathematics Subject Classification ID: 65N38
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 5009642
zbMATH Keywords: high dimensional integrals
zbMATH Keywords: exponential sums
zbMATH Keywords: integral equation method
zbMATH Keywords: numerical results
zbMATH Keywords: error estimates
zbMATH Keywords: approximation by exponentials
zbMATH Keywords: wavelets
reviewed by: D. Braess
MaRDI profile type: Publication
full work available at URL: s00607-005-0134-2
OpenAlex ID: W1988395629
cites work: Approximation of 1/x by exponential sums in [1, ∞)
cites work: The efficient computation of certain determinants arising in the treatment of Schrödinger's equations
DOI: 10.1007/S00607-005-0134-2
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description: scientific article; zbMATH DE number 5009642