Approximation of multidimensional functions with a given majorant of mixed moduli of continuity
type: Item
instance of: scholarly article
title: Approximation of multidimensional functions with a given majorant of mixed moduli of continuity
zbMATH Open document ID: 0949.42018
DOI: 10.1007/BF02675013
author: N. N. Pustovoĭtov
published in: Mathematical Notes
publication date: 2000-11-29T00:00:00Z
review text: Denote by \(\pi_d\) the cube \([-\pi,\pi]^d\), \(d\in \mathbb N\) and for \(q\geq 1\) let \(L_q^0(\pi_d)\) be the subspace of functions in \(L_q(\pi_d)\) for which \(\int_{-\pi}^{\pi}f(x) dx_i=0\), \(i=1,\dots,d.\) The classes \(H_q^\Omega\) of functions in \(L_q^0(\pi_d)\) are classes for which \(\Omega(f,t)_q=\sup\{\|\Delta_h^\ell f(x)\|_q\), \(|h_i|\leq t_i\), \(i=1,\dots,d\}\leq \Omega(t),\) where \(\Omega\) is a given function of the mixed \(\ell\)th-order modulus of continuity type. The author approximates the functions \(f\in H_q^\Omega\) (in the \(p\)-norm, \(p\leq q\)) by trigonometric polynomials of the form \[ T(x)=\sum_{k\in Q(N)}c_k e^{i(k,x)},\quad x=(x_1,\dots,x_q), \] where \(N\) is a given natural number and \(Q(N)=Q(N,\Omega)\) is a convenable set of multiindices generated by a level surface of the function \(\Omega.\) Under some conditions on \(\Omega\) one obtains the exact order for the upper bounds of the best approximation.
Mathematics Subject Classification ID: 42B99
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 1407538
zbMATH Keywords: best approximation of functions by trigonometric polynomials
zbMATH Keywords: mixed modulus of continuity
zbMATH Keywords: Kolmogorov widths
zbMATH Keywords: exact order upper bounds
reviewed by: Ioan Şerb
MaRDI profile type: Publication
cites work: Q3027578
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full work available at URL: https://doi.org/10.1007/bf02675013
OpenAlex ID: W1981326115
description: scientific article; zbMATH DE number 1407538