Constructive description of finitely generated ideals in an algebra of functions analytic in an open disc and smooth in its closure
type: Item
instance of: scholarly article
title: Constructive description of finitely generated ideals in an algebra of functions analytic in an open disc and smooth in its closure
zbMATH Open document ID: 0832.30023
author: F. A. Shamoyan
published in: Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences
publication date: 1995-12-12T00:00:00Z
review text: Denoting by \(\Lambda^a_\omega\) the class of holomorphic in the open unit disc \(D\) functions \(f\), for which \[ \bigl |f (\xi_1) - f (\xi_2) \bigr |\leq C_f \omega \bigl( |\xi_1 - \xi_2 |\bigr), \quad \xi_1, \xi_2 \in \overline D, \] where \(\omega\) is a nonnegative, increasing function satisfying Zygmund condition. The author shows that if \(g\), \(f_1, \ldots, f_n\) \[ \Lambda^a_\omega \text{ and } \bigl |g (\xi) \bigr |\leq C \biggl( \bigl |f_1 (\xi) \bigr |^2 + \cdots + \bigl |f_n (\xi) \bigr |^2 \biggr),\;\xi \in D, \] then there exist functions \(h_1, \ldots, h_n \in \Lambda^a_\omega\) such that \(g^2 (\xi) = f_1 (\xi) h_1 (\xi) + \cdots + f_n (\xi) h_n (\xi)\), \(\xi \in D\). He also shows that the converse is true in a sense. Moreover he considers the case of algebra \(\Lambda^{(m)}_\alpha\), \(0 < \alpha < 1\), \(m \geq 1\).
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 795474
zbMATH Keywords: corona problem
zbMATH Keywords: Zygmund condition
reviewed by: Takahiko Nakazi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 795474