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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Products of polynomials in uniform norms


Author: Igor E. Pritsker
Journal: Trans. Amer. Math. Soc. 353 (2001), 3971-3993
MSC (1991): Primary 30C10, 11C08, 30C15; Secondary 31A05, 31A15
Published electronically: May 21, 2001
MathSciNet review: 1837216
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Abstract:

We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. This subject includes the well known Gel'fond-Mahler inequalities for the unit disk and Kneser inequality for the segment $[-1,1]$. Using tools of complex analysis and potential theory, we prove a sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set of positive logarithmic capacity in the complex plane. The above classical results are contained in our theorem as special cases.

It is shown that the asymptotically extremal sequences of polynomials, for which this inequality becomes an asymptotic equality, are characterized by their asymptotically uniform zero distributions. We also relate asymptotically extremal polynomials to the classical polynomials with asymptotically minimal norms.


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Additional Information

Igor E. Pritsker
Affiliation: Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, Stillwater, Oklahoma 74078-1058
Email: igor@math.okstate.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02856-2
PII: S 0002-9947(01)02856-2
Keywords: Polynomials, uniform norm, zero counting measure, logarithmic capacity, equilibrium measure, subharmonic function, Fekete points
Received by editor(s): December 27, 1997
Published electronically: May 21, 2001
Additional Notes: Research supported in part by the National Science Foundation grants DMS-9996410 and DMS-9707359.
Article copyright: © Copyright 2001 American Mathematical Society