Computational Methods and Function Theory 5 (2005), No. 1, 135--141
Copyright Heldermann Verlag 2005
Mario Götz
mario.goetz@ku-eichstaett.de , Fachbereich Mathematik, Katholische Universität Eichstätt, 85071 Eichstätt, Germany.
For a bounded domain $G$ in the complex plane, we focus on the problem of maximizing the minimum on the boundary $\partial G$ of (weighted) polynomials of degree $n$ having all zeros in a set $D \subset G$. For arbitrary unit measures $\mu$ on $\partial G$ and weight $w := \exp\{U^\mu\}$, the $n$-th root asymptotics of $$ \sup_{p_n} \inf_{z \in \partial G} |p_n(z) w^n(z)| $$ is considered and related to the existence and construction of an inverse balayage of $\mu$ on $\overline{D}$, i.e.\ of a measure such that $\mu$ is its balayage when sweeping to $\partial G$.
Keywords: Logarithmic potential, weighted polynomial, equilibrium distribution, capacity, balayage, inverse balayage.
MSC 2000: 31A15, 30C85, 41A17.
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