Approximation by elements of a finite-dimensional subspace of functions from various Sobolev or Nikol'skij spaces
type: Item
instance of: scholarly article
title: Approximation by elements of a finite-dimensional subspace of functions from various Sobolev or Nikol'skij spaces
zbMATH Open document ID: 0708.41018
DOI: 10.1007/BF01158514
published in: Mathematical Notes
publication date: 1988-01-01T00:00:00Z
review text: Let \(F^ R_ q\) stand either for the anisotropic Sobolev class or for the Nikolskij class \((SW^ R_{q,\alpha}\) or \(NH^ R_ q)\). The following problem is treated: Select a proper ordering of a trigonometric system \(\{e^{i(k,x)}\}\) in order that the partial sums of periodic functions should give good approximation of these functions. The quantity \[ d_ M^{\perp}(F^ R_ q)_ p=\inf_{\{u_ i\}^ M_{i=1}}\sup_{f\in F^ R_ q}\| f-\sum^{M}_{i=1}(f,u_ i)\cdot u_ i\|_ p, \] where the infimum is taken over all orthonormal systems of bounded functions, is shown to satisfy certain estimates. Given M natural and a parallelepiped P, the author finds a system \(\{u_ i\}^ N_{i=1}\) with the least possible N such that for all \(R\in P\) the inequality \[ \sup_{f\in F^ R_ q}\| f- \sum^{N}_{i=1}(f,u_ i)u_ i\|_ p\leq K.d_ M^{\perp}(F^ R_ q)_ p \] holds. An asymptotical estimate for this smallest N by means of M is given.
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 4163458
zbMATH Keywords: anisotropic Sobolev class
zbMATH Keywords: Nikolskij class
zbMATH Keywords: orthonormal systems of bounded functions
zbMATH Keywords: asymptotical estimate
author: V. N. Temlyakov
reviewed by: Luboš Pick
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4163458