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Transactions of the American Mathematical Society

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

 

 

Capacity convergence results and applications
to a Berstein-Markov inequality


Authors: T. Bloom and N. Levenberg
Journal: Trans. Amer. Math. Soc. 351 (1999), 4753-4767
MSC (1991): Primary 31C15, 32F05, 41A17
Published electronically: August 25, 1999
MathSciNet review: 1695017
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Abstract | References | Similar Articles | Additional Information

Abstract: Given a sequence $\{E_{j}\}$ of Borel subsets of a given non-pluripolar Borel set $E$ in the unit ball $B$ in $\mathbf{C}^{N}$ with $E \subset \subset B$, we show that the relative capacities $C(E_{j})$ converge to $C(E)$ if and only if the relative (global) extremal functions $u_{E_{j}}^{*}$ ($V_{E_{j}}^{*}$) converge pointwise to $u_{E}^{*}$ ($V_{E}^{*}$). This is used to prove a sufficient mass-density condition on a finite positive Borel measure with compact support $K$ in $\mathbf{C}^{N}$ guaranteeing that the pair $(K,\mu )$ satisfy a Bernstein-Markov inequality. This implies that the $L^{2}-$orthonormal polynomials associated to $\mu $ may be used to recover the global extremal function $V_{K}^{*}$.


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Additional Information

T. Bloom
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada
Email: bloom@math.toronto.edu

N. Levenberg
Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand
Email: levenber@math.auckland.ac.nz

DOI: https://doi.org/10.1090/S0002-9947-99-02556-8
Received by editor(s): February 11, 1998
Received by editor(s) in revised form: March 5, 1999
Published electronically: August 25, 1999
Article copyright: © Copyright 1999 American Mathematical Society