A Bernstein-type inequality for rational functions in weighted Bergman spaces

2010/03/26 by Baranov, Anton, Zarouf, Rachid
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1003.5066

Abstract

Given n≥1 and r∈[0, 1), we consider the set R_{n, r} of rational functions having at most n poles all outside of (1)/(r)\mathbbD, were \mathbbD is the unit disc of the complex plane. We give an asymptotically sharp Bernstein-type inequality for functions in R_{n, r} (as n tends to infinity and r tends to 1-) in weighted Bergman spaces with "polynomially" decreasing weights. We also prove that this result can not be extended to weighted Bergman spaces with "super-polynomially" decreasing weights.

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