Asymptotic behavior of the Gelfand and the Bernstein widths of some classes of analytic functions
type: Item
instance of: scholarly article
title: Asymptotic behavior of the Gelfand and the Bernstein widths of some classes of analytic functions
zbMATH Open document ID: 0880.46040
DOI: 10.1007/BF02509564
author: O. G. Parfenov
publication date: 1998-01-07T00:00:00Z
review text: Let \(H^p\), \(1\leq p\leq\infty\), be the Hardy class in the unit disk \(\mathbb{D}\), and let \(T_r\) be the circle \(\{z:|z|=r\}\), \(0<r\leq 1\), \(\mathbb{T}= T_1\). The author studied the Gelfand widths of the unit ball \(BH^p\) in the metric of the space \(L_q(T_r,\mu)\) for \(p\geq q\) and for an arbitrary measure \(\mu\) on \(T_r\). Fisher and Stessin proved that for \(p\geq q\) the Kolmogorov, the Gelfand, and the linear widths of \(BH^p\) in the space \(L_q(K,\mu)\) (when \(K\) is a compact set in \(\mathbb{D}\)) coincide. For \(p<q\), very little is known about the widths in question. The author established Ismagilov type theorems for the Gelfand and the Bernstein widths. From the results of [the author, Mat. Sb., n. Ser. 117(159), No. 2, 279-285 (1982; Zbl 0488.30027) and Zap. Nauchn. Sem. 217, 112-129 (1994)] one can obtain nontrivial lower bounds for the Gelfand widths and upper bounds for the Bernstein widths. However, it is difficult to derive the reverse estimates from these results. The present paper deals with just this problem.
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 1002988
zbMATH Keywords: Hardy class in the unit disk
zbMATH Keywords: Gelfand widths of the unit ball
zbMATH Keywords: Ismagilov type theorems
zbMATH Keywords: Bernstein widths
zbMATH Keywords: lower bounds for the Gelfand widths
zbMATH Keywords: upper bounds for the Bernstein widths
MaRDI profile type: Publication
cites work: Q3247378
full work available at URL: https://doi.org/10.1007/bf02509564
OpenAlex ID: W2057427330
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1002988