Products of polynomials in uniform norms

Pritsker, Igor

doi:10.1090/S0002-9947-01-02856-2

Products of polynomials in uniform norms
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by Igor E. Pritsker

Trans. Amer. Math. Soc. 353 (2001), 3971-3993

DOI: https://doi.org/10.1090/S0002-9947-01-02856-2

Published electronically: May 21, 2001

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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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