The Jackson inequality and widths of function classes in \(L^2([0,1],x^{2v+1})\)
type: Item
instance of: scholarly article
title: The Jackson inequality and widths of function classes in \(L^2([0,1],x^{2v+1})\)
zbMATH Open document ID: 1258.41006
author: Chun-Mei Su
author: Yong-ping Liu
author: Jian Li
published in: Journal of Complexity
publication date: 2012-10-10T00:00:00Z
reviewed by: Yuri A. Farkov
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 6092444
zbMATH Keywords: Jackson inequality
zbMATH Keywords: continuous modulus of order \(\alpha \)
zbMATH Keywords: derivative of order \(\lambda \)
zbMATH Keywords: Bessel function
zbMATH Keywords: \(n\)-widths of function classes
MaRDI profile type: Publication
full work available at URL: j.jco.2012.04.004
OpenAlex ID: W159761946
cites work: On Jackson's inequality in \(L_2\)
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cites work: Jackson-type inequalities and widths of function classes in \(L_{2}\)
DOI: 10.1016/J.JCO.2012.04.004
review text: Let \(J_{\nu}(x)\) be the Bessel function of order \(\nu>-1/2\) and let \(\{\mu_l^{(\nu)}\}\) be the set of all positive roots of \(J_{\nu}(x)\), which are numbered such that \(0< \mu_l^{(\nu)} < \mu_{l+1}^{(\nu)}\) for all \(l\in {\mathbb N}\). Then for \(f\in L^2([0,1], x^{2\nu +1})\) we have the expansion NEWLINE\[NEWLINE f(x)=\sum_{l=1}^{\infty}a_l\varphi_l(x), \quad a_l=\frac{(f,\varphi_l)}{\varphi_l,\varphi_l}, \eqno (1) NEWLINE\]NEWLINE where \(\varphi_l(x)=j_{\nu}(\mu_l^{(\nu)}x)\) with \(j_{\nu}(x)=2^{\nu}\Gamma(\nu + 1)J_{\nu}(x)/x^{\nu}\) for \(x\neq 0\) and \(j_{\nu}(0)=1\). Using (1), the authors define the (generalized) derivative \(\widetilde D^{(\lambda)}f\) and the continuous modulus \(\omega_{\alpha}(f,\tau)\), where \(\lambda\in {\mathbb R}_+\), \(\alpha, \, \tau > 0\). We write \(f\in L^{2,\lambda}([0,1], x^{2\nu +1})\) if the derivative \(\widetilde D^{(\lambda)}f\) exists and belongs to the space \(L^2([0,1], x^{2\nu +1})\). Further, let \(E_n(f,{\mathcal L}_n)\) denote the best approximation of \(f\) by the subspace \({\mathcal L}_n=\text{span}\,\{\varphi_1, \dots, \varphi_n\}\) in the metric \(L^2([0,1], x^{2\nu +1})\). For small enough \(t\), the problem on the sharp Jackson constant \(K_{n,\alpha,\lambda}(t)\) in the inequality NEWLINE\[NEWLINE E_n(f,{\mathcal L}_n)\leq K_{n,\alpha,\lambda}(t)(\mu_{n+1}^{(\nu)})^{-\lambda} \omega_{\alpha}(\widetilde D^{(\lambda)}f,t), \quad f\in L^{2,\lambda}([0,1], x^{2\nu +1}), NEWLINE\]NEWLINE is solved. Moreover, the exact values of the \(n\)-widths of the classes \(S(\alpha, \lambda,\Psi)\) and \(\mathfrak{S}(\alpha, \lambda)\) in the space \(L^2([0,1], x^{2\nu +1})\) are found.
description: scientific article; zbMATH DE number 6092444