Optimal recovery of periodic functions from Fourier coefficients given with an error
type: Item
instance of: scholarly article
title: Optimal recovery of periodic functions from Fourier coefficients given with an error
zbMATH Open document ID: 0859.42004
published in: Journal of Complexity
publication date: 1996-06-04T00:00:00Z
review text: The paper is concerned with optimal recovery of linear functionals on Hilbert spaces based on approximate Fourier coefficients with a prescribed error in the uniform norm. The general framework is the following: Let \(X\) be a Hilbert space with a complete orthonormal system \((e_j,j\in\mathbb{N})\) and let \(x_j=(x,e_j)\), \(j\in\mathbb{N}\), denote the Fourier coefficients of \(x\). Let also \(B_X\) be the closed unit ball of \(X\). For \(f\in X\) an optimal recovery of the functional \((x,f)\), \(x\in B_X\), from approximate Fourier coefficients \(\widetilde x_j\) verifying \(|x_j-\widetilde x_j|\leq \delta_j\), \(j=1,\dots, n\), is given by \(\sum^n_{j=1} (1-\lambda\delta_j|f_j|^{-1})_+\overline f_j\widetilde x_j\), where \(\lambda\in(0,|f|)\) is the solution of the equation \(|f|^2- \sum^n_{j=1} (|f_j|^2-\lambda^2\delta^2_j)_+-\lambda^2=0\). (Here, as usual, \(a_+=2^{-1}(|a|+a))\). The optimal value of the method is \(e(f,\delta)=\lambda+ \sum^n_{j=1} \delta_j(|f_j|-\lambda\delta_j)_+\) (Theorem 1). The obtained results are applied to optimal recovery of \(2\pi\)-periodic functions and of their derivatives in Hardy-Sobolev and Bergman-Sobolev spaces. A similar problem, but for exact values of the Fourier coefficients, was solved by \textit{B. D. Boyanov} [Serdica 2, 300-304 (1976; Zbl 0442.42005)].
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 65Y20
zbMATH DE Number: 884756
zbMATH Keywords: optimal recovery
zbMATH Keywords: linear functionals on Hilbert spaces
zbMATH Keywords: approximate Fourier coefficients
zbMATH Keywords: Hardy-Sobolev and Bergman-Sobolev spaces
author: K. Yu. Osipenko
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: jcom.1996.0005
OpenAlex ID: W2156919634
DOI: 10.1006/JCOM.1996.0005
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description: scientific article; zbMATH DE number 884756