# Maximin Polynomials and Inverse Balayage

Maximin Polynomials and Inverse Balayage For a bounded domain G in the complex plane, we focus on the problem of maximizing the minimum on the boundary ∂G of (weighted) polynomials of degree n having all zeros in a set D ⊂ G. For arbitrary unit measures μ on ∂ G and weight w:= exp{Uμ}, the n-th root asymptotics of \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\matrix{{\rm sup} \cr pn} \matrix{{\rm inf} \cr z \varepsilon \partial G} \mid P_n(z)\omega^n(z) \mid$$\end{document} is considered and related to the existence and construction of an inverse balayage of μ on \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline D$$\end{document}, i.e. of a measure such that μ is its balayage when sweeping to ∂G. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# Maximin Polynomials and Inverse Balayage

, Volume 5 (1) – Aug 1, 2005
7 pages      /lp/springer-journals/maximin-polynomials-and-inverse-balayage-LrC0e8n8KL
Publisher
Springer Journals
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/bf03321090
Publisher site
See Article on Publisher Site

### Abstract

For a bounded domain G in the complex plane, we focus on the problem of maximizing the minimum on the boundary ∂G of (weighted) polynomials of degree n having all zeros in a set D ⊂ G. For arbitrary unit measures μ on ∂ G and weight w:= exp{Uμ}, the n-th root asymptotics of \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\matrix{{\rm sup} \cr pn} \matrix{{\rm inf} \cr z \varepsilon \partial G} \mid P_n(z)\omega^n(z) \mid$$\end{document} is considered and related to the existence and construction of an inverse balayage of μ on \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline D$$\end{document}, i.e. of a measure such that μ is its balayage when sweeping to ∂G.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: Aug 1, 2005

Keywords: Logarithmic potential; weighted polynomial; equilibrium distribution; capacity; balayage; inverse balayage; 31A15; 30C85; 41A17

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