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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

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Publisher
Springer Journals
Copyright
Copyright © Heldermann  Verlag 2005
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 Theory – Springer Journals

Published: Aug 1, 2005

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

References

  • On polynomials most divergent from zero on a domain boundary
    Borodin, P A
  • Über Tschebyscheffsche Polynome
    Faber, G
  • Potential and discrepancy estimates for weighted extremal points
    Götz, M; Saff, E B
  • Direct and inverse balayage — some new developments in classical potential theory
    Gustafsson, B
  • On mother bodies of convex polyhedra
    Gustafsson, B