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On Bernstein Inequality via Chebyshev Polynomial

On Bernstein Inequality via Chebyshev Polynomial Motivated by applications to the Carleson embedding theorem with matrix weights, Culiuc and Treil proved a Bernstein-type inequality for complex polynomials in the plane which are positive and satisfy a polynomial growth condition on the positive real axis. A sharp form of this Bernstein inequality, with Chebyshev polynomial of the first kind as an extremizer, was later found by Kraus, Moucha and Roth. In this note we show that the Chebyshev polynomial of the first kind is indeed the only extremal polynomial for this sharp Bernstein inequality. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

On Bernstein Inequality via Chebyshev Polynomial

Computational Methods and Function Theory , Volume OnlineFirst – May 23, 2022

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-022-00454-4
Publisher site
See Article on Publisher Site

Abstract

Motivated by applications to the Carleson embedding theorem with matrix weights, Culiuc and Treil proved a Bernstein-type inequality for complex polynomials in the plane which are positive and satisfy a polynomial growth condition on the positive real axis. A sharp form of this Bernstein inequality, with Chebyshev polynomial of the first kind as an extremizer, was later found by Kraus, Moucha and Roth. In this note we show that the Chebyshev polynomial of the first kind is indeed the only extremal polynomial for this sharp Bernstein inequality.

Journal

Computational Methods and Function TheorySpringer Journals

Published: May 23, 2022

Keywords: Bernstein inequalities; Chebyshev polynomials; Carleson embedding; Primary 41A17; Secondary 30C10

References