Gleason's problem and homogeneous interpolation in Hardy and Dirichlet-type spaces of the ball
type: Item
instance of: scholarly article
title: Gleason's problem and homogeneous interpolation in Hardy and Dirichlet-type spaces of the ball
zbMATH Open document ID: 1022.47010
DOI: 10.1016/S0022-247X(02)00412-2
author: Daniel Alpay
author: H. Turgay Kaptanoğlu
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: The authors consider the reproducing kernel Hilbert spaces \({\mathcal H}_r\) with reproducing kernels \((1-\langle z,w\rangle)^{-r}\) for \(r>0\) and \(\log (1-\langle z,w\rangle)^{-1}\) for \(r=0\) on the unit ball of \({\mathbb C}^N\). These are spaces which are often called Dirichlet-type spaces or weighted Hardy (Bergman) spaces. The authors solve Gleason's problem for \({\mathcal H}_r\) by using the method of backward shift operators. Their proof actually works on any reproducing kernel Hilbert space with reproducing kernels which are holomorphic in \(\langle z,w\rangle\). By using these results, the authors obtain a description of the set of solutions of certain finite homogeneous interpolation problems in spaces of the form \({\mathcal H}_r^{n\times 1}\). These results extend the previous work of the authors for \({\mathcal H}_1\) [Integral Equations Oper. Theory 42, 1--21 (2001; Zbl 1010.47012)] to general \(r\).
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 32E30
Mathematics Subject Classification ID: 46E22
zbMATH DE Number: 1866770
zbMATH Keywords: reproducing kernel Hilbert space
zbMATH Keywords: Dirichlet-type space
zbMATH Keywords: Gleason's problem
zbMATH Keywords: homogeneous interpolation
reviewed by: Boo Rim Choe
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1866770