Gel'fand widths of the unit ball of the Hardy class \(H^ p\) in weight spaces
type: Item
instance of: scholarly article
title: Gel'fand widths of the unit ball of the Hardy class \(H^ p\) in weight spaces
zbMATH Open document ID: 0592.46050
DOI: 10.1007/BF01156751
author: O. G. Parfenov
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: Let \(H^ p\), \(1\leq p\leq \infty\), be the classical Hardy class of order p, \(B_ p\) the unit disk in \(H^ p\), \(T_ r=[| z| =r]\), \(\mu\) a finite Borel measure on \(T_ r\) and \(L_ q(T_ r,\mu)\), \(1\leq q\leq \infty\), the usual space of Lebesgue integrable functions of order q. Suppose that \(d^ n=d^ n(B_ p(T_ r,\mu))\) denotes the n-th diameter in the sense of Gel'fand between the two indicated sets. The author proved that \[ \lim_{n\to \infty}d^ nr^{1-n- (1/q)}=(2\pi)^{(1/q)-(1/p)}(g(\mu))^{1/q}, \] where \(g(\mu)=\exp [(2\pi r)^{-1}\int_{T_ r}\ln p(z)| dz|]\).
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 41A46
zbMATH DE Number: 3951319
zbMATH Keywords: n-th diameter in the sense of Gel'fand
reviewed by: Guillermo López Lagomasino
MaRDI profile type: Publication
cites work: Q5527180
cites work: The n-width of sets of analytic functions
cites work: Q3935478
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description: scientific article; zbMATH DE number 3951319