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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Products of polynomials in uniform norms
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by Igor E. Pritsker PDF
Trans. Amer. Math. Soc. 353 (2001), 3971-3993 Request permission

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
  • MR Author ID: 319712
  • Email: igor@math.okstate.edu
  • 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.
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 3971-3993
  • MSC (1991): Primary 30C10, 11C08, 30C15; Secondary 31A05, 31A15
  • DOI: https://doi.org/10.1090/S0002-9947-01-02856-2
  • MathSciNet review: 1837216