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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Polynomials with zeros and small norm on curves


Author: Vilmos Totik
Journal: Proc. Amer. Math. Soc. 140 (2012), 3531-3539
MSC (2010): Primary 41A10, 31A15
Published electronically: February 23, 2012
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Abstract: This paper considers the problem of how zeros lying on the boundary of a domain influence the norm of polynomials (under the normalization that their value is fixed at a point). It is shown that $ k$ zeros raise the norm by a factor $ (1+ck/n)$ (where $ n$ is the degree of the polynomial), while $ k$ excessive zeros on an arc compared to $ n$ times the equilibrium measure raise the norm by a factor $ \exp (ck^2/n)$. These bounds are sharp, and they generalize earlier results for the unit circle which are connected to some constructions in number theory. Some related theorems of Andrievskii and Blatt will also be strengthened.


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Additional Information

Vilmos Totik
Affiliation: Bolyai Institute, Analysis Research Group of the Hungarian Academy of Sciences, University of Szeged, Szeged, Aradi v. tere 1, 6720, Hungary – and – Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Avenue, PHY 114, Tampa, Florida 33620-5700
Email: totik@mail.usf.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11223-7
PII: S 0002-9939(2012)11223-7
Keywords: Polynomials, zeros, small supremum norm
Received by editor(s): April 12, 2011
Published electronically: February 23, 2012
Additional Notes: The author was supported by ERC grant No. 267055
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2012 American Mathematical Society