Asymptotic behavior of the singular numbers of the imbedding operator of the Hardy class \(H^ 2\) into the weight space \(L_ 2(D_ r,\mu)\)
type: Item
instance of: scholarly article
title: Asymptotic behavior of the singular numbers of the imbedding operator of the Hardy class \(H^ 2\) into the weight space \(L_ 2(D_ r,\mu)\)
zbMATH Open document ID: 0577.42020
DOI: 10.1007/BF01137697
author: O. G. Parfenov
published in: Mathematical Notes
publication date: 1984-01-01T00:00:00Z
review text: Let \(H^ 2\) be a Hardy class in the unit circle D. We denote \(D_ r=\{z:\) \(| z| <r\}\) and \(T_ r\) is the boundary of \(D_ r\). Let \(\mu\) be an arbitrary measure on \(\bar D_ r\) and \(\mu_ 0\) its restriction on \(T_ r\). In this note it is proved that the singular numbers \(\{s_ n\}\) of the imbedding operator I of \(H^ 2\) into \(L_ 2(\bar D_ r,\mu)\) satisfies the relation \(\lim_{n\to \infty} s_ n(I)r^{1/2-n}=\sqrt{g(\mu_ 0)},\) where \(g(\mu_ 0)=\exp [(2\pi r)^{-1}\int_{T_ r}n((d\mu_ 0/| dz|)| dz|].\)
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 3923362
zbMATH Keywords: restriction of measure
zbMATH Keywords: Hardy class
zbMATH Keywords: imbedding operator
MaRDI profile type: Publication
cites work: Q3949270
cites work: The n-width of sets of analytic functions
description: scientific article; zbMATH DE number 3923362