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An extremal problem for characteristic functions


Authors: Isabelle Chalendar, Stephan Ramon Garcia, William T. Ross and Dan Timotin
Journal: Trans. Amer. Math. Soc. 368 (2016), 4115-4135
MSC (2010): Primary 47A05, 47B35, 47B99
DOI: https://doi.org/10.1090/tran/6446
Published electronically: September 2, 2015
MathSciNet review: 3453366
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Abstract: Suppose $ E$ is a subset of the unit circle $ \mathbb{T}$ and $ H^\infty \subset L^\infty $ is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of $ E$ to $ z^nH^\infty $. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.


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Additional Information

Isabelle Chalendar
Affiliation: Institut Camille Jordan, University of Lyon I, 43 Boulevard du 11 Bovembre 1918, 69622 Villeurbanne cedex, France
Email: chalendar@math.univ-lyon1.fr

Stephan Ramon Garcia
Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
Email: Stephan.Garcia@pomona.edu

William T. Ross
Affiliation: Department of Mathematics and Computer Science, University of Richmond, Richmond, Virginia 23173
Email: wross@richmond.edu

Dan Timotin
Affiliation: Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 014700, Romania
Email: Dan.Timotin@imar.ro

DOI: https://doi.org/10.1090/tran/6446
Keywords: Extremal problem, truncated Toeplitz operator, Toeplitz operator, Hankel operator, complex symmetric operator
Received by editor(s): July 10, 2013
Received by editor(s) in revised form: April 7, 2014
Published electronically: September 2, 2015
Additional Notes: The second author was partially supported by National Science Foundation Grant DMS-1001614
The fourth author was partially supported by a grant of the Romanian National Authority for Scientific Research, CNCS Ð UEFISCDI, project number PN-II-ID-PCE-2011-3-0119.
Article copyright: © Copyright 2015 American Mathematical Society