Some Results on the Rational Bernstein–Markov Property in the Complex Plane

The Bernstein–Markov property is an asymptotic quantitative assumption on the growth of uniform norms of polynomials or rational functions on a compact set with respect to L μ 2 -norms, where μ is a positive finite measure. We consider two variants of the Bernstein–Markov property for rational funct...

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Bibliographic Details
Published in:Computational methods and function theory 2017-09, Vol.17 (3), p.405-443
Main Author: Piazzon, Federico
Format: Article
Language:English
Subjects:
Analysis
Asymptotic properties
Computational Mathematics and Numerical Analysis
Functions (mathematics)
Functions of a Complex Variable
Markov processes
Mathematical analysis
Mathematics
Mathematics and Statistics
Norms
Polynomials
Rational functions
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Summary:The Bernstein–Markov property is an asymptotic quantitative assumption on the growth of uniform norms of polynomials or rational functions on a compact set with respect to L μ 2 -norms, where μ is a positive finite measure. We consider two variants of the Bernstein–Markov property for rational functions with restricted poles and compare them with the polynomial Bernstein–Markov property to find some sufficient conditions for the latter to imply the former. Moreover, we recover a sufficient mass-density condition for a measure to satisfy the rational Bernstein–Markov property on its support. Finally we present, as an application, a meromorphic L 2 version of the Bernstein–Walsh Lemma.
ISSN:1617-9447
2195-3724
DOI:10.1007/s40315-017-0194-2