Extremal functions in invariant subspaces of Bergman spaces
type: Item
instance of: scholarly article
title: Extremal functions in invariant subspaces of Bergman spaces
zbMATH Open document ID: 0853.30023
author: Peter Duren
author: H. S. Shapiro
author: Dmitry Khavinson
published in: Illinois Journal of Mathematics
publication date: 1996-12-11T00:00:00Z
review text: Some recent results on contractive zero-divisors in the Bergman spaces \(A^p\) \((0 < p < \infty)\) are partially generalized to arbitrary subspaces invariant under multiplication by polynomials. It is shown that each invariant subspace of \(A^p\) generated by an inner function has a unique extremal function, represented in terms of the reproducing kernel of a certain weighted \(A^2\) space. The extremal function is shown to have an analytic continuation across each arc of the unit circle that lies outside the singular set of the generator. It is also shown that the canonical divisor in \(A^p\) of a Blaschke sequence generates the invariant subspace determined by the zero-set, as does the associated Blaschke product. Thus the canonical divisor of a Blaschke sequence has an analytic continuation over each arc of the unit circle containing no cluster points of the zeros.
reviewed by: Peter Duren
Mathematics Subject Classification ID: 30D99
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 926190
zbMATH Keywords: Bergman spaces
zbMATH Keywords: analytic continuation
zbMATH Keywords: inner function
zbMATH Keywords: extremal function
zbMATH Keywords: canonical divisor
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 926190