Determination of inner functions by their value sets on the circle
type: Item
instance of: scholarly article
title: Determination of inner functions by their value sets on the circle
zbMATH Open document ID: 1271.30031
DOI: 10.1007/BF03321808
author: Isabelle Chalendar
author: Pamela Gorkin
author: Jonathan R. Partington
published in: Computational Methods and Function Theory
publication date: 2011-10-28T00:00:00Z
review text: This paper is devoted to the question of reconstructing an inner function from its values at a discrete set of points on the unit circle. More precisely, by establishing an explicit formula, the authors show that the set of points at which such a function takes two values of modulus one determines an inner function. If the inner function has only one singularity, it is unique up to a hyperbolic automorphism. They also obtain a full characterization of the sets at which the prescribed values can be taken. The authors first study interpolation by finite Blaschke products where each unimodular value is taken finitely many times on the circle. They link this question with the study of functions mapping the upper half-plane into itself and the real axis to the extended real axis. They make heavy use of the Hermite-Biehler criterion (also attributed to M. G. Krein) which characterizes the real meromorphic functions (that is, those meromorphic functions in the plane which take real values or the infinite value on the real line) mapping the upper half-plane to itself. The paper contains relevant examples, as well as applications to isometric composition operators and to the numerical range of the restricted shift operator on the model space of an inner function.
reviewed by: Dragan Vukotić
Mathematics Subject Classification ID: 30J05
Mathematics Subject Classification ID: 30H10
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47A12
zbMATH DE Number: 5965046
zbMATH Keywords: inner functions
zbMATH Keywords: Blaschke products
zbMATH Keywords: real meromorphic functions
zbMATH Keywords: composition operators
zbMATH Keywords: model space of an inner function
zbMATH Keywords: numerical range
MaRDI profile type: Publication
Wikidata QID: Q56319193
cites work: Numerical ranges of \(C_{0}(N)\) contractions
cites work: Q3643410
cites work: On the structure of invariant subspaces for isometric composition operators on \(H^2(\mathbb{D})\) and \(H^2(\mathbb{C}_+)\)
cites work: One dimensional perturbations of restricted shifts
cites work: Similarity to an isometry of composition operators on the half-plane
cites work: Numerical range ofs(φ)
cites work: Numerical range and Poncelet property.
cites work: Über den numerischen Wertebereich eines Operators
cites work: Q5727906
cites work: Q5531193
cites work: Q4105628
cites work: Q4787437
cites work: On a Connection Between the Numerical Range and Spectrum of an Operator on a Hilbert Space
cites work: Eigenvalues in the boundary of the numerical range
full work available at URL: https://doi.org/10.1007/bf03321808
OpenAlex ID: W2001528885
description: scientific article; zbMATH DE number 5965046