Applications of Smolyak quadrature formulas to the numerical integration of Fourier coefficients and in function recovery problems
type: Item
instance of: scholarly article
title: Applications of Smolyak quadrature formulas to the numerical integration of Fourier coefficients and in function recovery problems
zbMATH Open document ID: 1205.65346
DOI: 10.3103/S1066369X10030084
author: S. S. Kudaibergenov
author: A. A. Shomanova
author: Nurlan Temirgaliyev
published in: Russian Mathematics
publication date: 2010-09-23T00:00:00Z
review text: Let \(f: {\mathbb R}^s \to {\mathbb C}\) be a sufficiently smooth function which is 1-periodic in each variable. Using the Smolyak quadrature formula, which uses only nodes of a sparse grid in \([0,\,1]^s\), see \textit{S. A. Smolyak} [Dokl. Akad. Nauk SSSR 148, 1042--1045 (1963; Zbl 0202.39901)], the authors calculate the Fourier coefficients \[ {\hat f}(n) = \int_{[0,1]^s} f(x)\, e^{-2\pi i \,(n,x)}\, dx \quad (n\in {\mathbb Z}^s). \] If \(f\) belongs to generalized Sobolev or Korobov space, the exact orders of the errors of this numerical integration are determined by the authors. Finally, these results are applied to an approximate recovery of \(f\), where the exact Fourier coefficients \({\hat f}(n)\) of a finite Fourier sum are replaced by their computed values.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 42B05
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 5790424
zbMATH Keywords: Smolyak quadrature formula
zbMATH Keywords: sparse grid
zbMATH Keywords: multivariate periodic function
zbMATH Keywords: computation of Fourier coefficients
zbMATH Keywords: approximate recovery
zbMATH Keywords: generalized Sobolev space
zbMATH Keywords: generalized Korobov space
zbMATH Keywords: function spaces of dominating mixed smoothness
zbMATH Keywords: finite Fourier sum
MaRDI profile type: Publication
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full work available at URL: s1066369x10030084
OpenAlex ID: W2026518668
description: scientific article; zbMATH DE number 5790424