\(L_ 2(\mathbb{R}^ n)\)-extremal problems on the basis of incomplete information
type: Item
instance of: scholarly article
title: \(L_ 2(\mathbb{R}^ n)\)-extremal problems on the basis of incomplete information
zbMATH Open document ID: 0859.41020
DOI: 10.1007/BF01904272
published in: Analysis Mathematica
publication date: 1995-11-26T00:00:00Z
review text: The paper under review deals with the best approximation problem for a densely defined, Banach space valued linear operator. Its purpose is to extend solutions of the best approximation problem from Sobolev spaces on the torus and the real line to the multivariate case by means of polyharmonic splines. The \(m\)th-polyharmonic splines on the space \(\mathbb{R}^n\) are annihilated by the \(m\)-th iterated Laplacian \(\Delta^m\) on the points of \(\mathbb{R}^n\), with exception of those points which belong to the lattice \(\mathbb{Z}^n\). For even numbers \(m\), they give rise to the cardinal interpolation problem on the space \(\mathbb{R}^n\). -- The paper is written in the technical vein characteristic for multivariate approximation, and the theory of function spaces on \(\mathbb{R}^n\). A constituent ingredient is formed by the Hardy-Littlewood-Pólya inequality which follows from the Plancherel formula in conjunction with Hölder's inequality.
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 808263
zbMATH Keywords: optimal recovery
zbMATH Keywords: Hardy-Littlewood-Pólya inequality
zbMATH Keywords: Plancherel formula
author: Yong-ping Liu
reviewed by: Walter Schempp
MaRDI profile type: Publication
cites work: Q4109656
cites work: Polyharmonic cardinal splines
cites work: Polyharmonic cardinal splines: A minimization property
cites work: FINITENESS OF $ \textit{Ø}$ OVER TOTALLY REAL FIELDS
cites work: Q3349384
cites work: Optimal recovery forW 2 r (ℝ) inL 2 (ℝ)
cites work: Q5607484
cites work: The Brauer-Manin Obstruction and III[2]
full work available at URL: https://doi.org/10.1007/bf01904272
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description: scientific article; zbMATH DE number 808263