Sharp inequalities for approximations of classes of periodic convolutions by odd-dimensional subspaces of shifts
type: Item
instance of: scholarly article
title: Sharp inequalities for approximations of classes of periodic convolutions by odd-dimensional subspaces of shifts
zbMATH Open document ID: 1211.41003
DOI: 10.1134/S0001434609030250
author: O. L. Vinogradov
published in: Mathematical Notes
publication date: 2009-11-10T00:00:00Z
review text: The paper is dedicated to obtain sharp inequalities of Favard-Akhiezer-Krein type for classes of periodic convolutions with kernels that do not increase oscillation. Let \(W_p^{(r)}\) denote the space of \(2\pi\)-periodic continuous functions \(f\) for which \(f^{(r-1)}\) is absolutely continuous and \(f^{(r)}\in L_p\). Let \({\mathcal T}_{2n-1}\) denote the space of trigonometric polynomials of order at most \(2n-1\). In 1937, Favard and, independently, Akhiezer and Krein constructed a linear approximation method \(X_{n,r}: L_1 \to {\mathcal T}_{2n-1}\) for \(f\in W_p^{(r)}\) (with \(p=\infty\)) such that \[ \|f-X_{n,r}(f)\|_p\leq \frac{{\mathcal K}_r}{n^r}\|f^{(r)}\|_{p}, \qquad p=\infty, \tag{1} \] where \({\mathcal K}_r\) are the in Approximation Theory celebrated Favard constants. This inequality is sharp. S. M. Nikol'skii later extended this result to the case \(p=1\). After that many mathematicians have been working on obtaining sharp results of this type for various classes of functions and various approximating methods. One of the natural ways to generalize this result is to consider classes of convolutions with certain kernels \(G\) (i.e., \(f=\varphi \star G\)) that do not increase oscillation. The author obtains sharp inequalities of the form (1) for approximations of classes of convolutions by subspaces constructed by uniform shifts of a single function. Conditions on the functions generating approximating subspaces are stated in terms of non increasing oscillations (see below for the precise conditions). Denote by \(A_{2n}(J)\) the set of real-valued functions \(G\in L_1\) satisfying the following condition on the number of sign changes \[ \nu(T+G\star \varphi) \leq 2n \] for any functions \(\varphi\) and \(T\) with the following properties: \(T\) is in the linear span of the set \(\{exp(ik(\cdot)): k\in J\}\) (denoted by \({\mathcal T}_J\)), \(\varphi\in L_1\), \(\varphi \perp {\mathcal T}_J\), and the number of essential changes of sign of the functi…
reviewed by: Yuliya Babenko
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 42A85
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 5628183
zbMATH Keywords: inequality
zbMATH Keywords: Favard-Akhiezer-Krein inequality
zbMATH Keywords: Jackson inequality
zbMATH Keywords: spline
zbMATH Keywords: periodic convolution
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5628183