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Remarks on Inner Functions and Optimal Approximants

Published online by Cambridge University Press:  20 November 2018

Catherine Bénéteau
Affiliation:
Department of Mathematics, University of South Florida, Tampa, FL 33620, USA, e-mail : cbenetea@usf.edu
Matthew C. Fleeman
Affiliation:
Baylor University, Waco, TX 76710, USA, e-mail : matthew_fleeman@baylor.edu
Dmitry S. Khavinson
Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA, e-mail : dkhavins@usf.edu
Daniel Seco
Affiliation:
Departament de Matemàtica Aplicada i Anàlisi, Facultat de Matemàtiques, Universitat de Barcelona, 08007 Barcelona, Spain, e-mail : dseco@mat.uab.cat
Alan A. Sola
Affiliation:
Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden, e-mail : sola@math.su.se

Abstract

We discuss the concept of inner function in reproducing kernelHilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to ${1}/{f}\;$ , where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modiûed to produce inner functions.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2018

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