Transfinite diameter, Chebyshev constants, and capacities in ℂⁿ

Vyacheslav Zakharyuta. "Transfinite diameter, Chebyshev constants, and capacities in ℂⁿ." Annales Polonici Mathematici 106.1 (2012): 293-313. <http://eudml.org/doc/280619>.

@article{VyacheslavZakharyuta2012,
abstract = {The famous result of geometric complex analysis, due to Fekete and Szegö, states that the transfinite diameter d(K), characterizing the asymptotic size of K, the Chebyshev constant τ(K), characterizing the minimal uniform deviation of a monic polynomial on K, and the capacity c(K), describing the asymptotic behavior of the Green function $g_\{K\}(z)$ at infinity, coincide. In this paper we give a survey of results on multidimensional notions of transfinite diameter, Chebyshev constants and capacities, related to these classical results and initiated by Leja’s definition of transfinite diameter of a compact set K⊂ ℂⁿ and the author’s paper [Mat. Sb. 25 (1975)], where a multidimensional analog of the Fekete equality d(K) = τ(K) was first considered for any compact set in ℂⁿ. Using some general approach, we introduce an alternative definition of transfinite diameter and show its coincidence with Fekete-Leja’s transfinite diameter. In conclusion we discuss an application of the results of the author’s paper mentioned above to the asymptotics of the leading coefficients of orthogonal polynomial bases in Hilbert spaces related to a given pluriregular polynomially convex compact set in ℂⁿ.},
author = {Vyacheslav Zakharyuta},
journal = {Annales Polonici Mathematici},
keywords = {transfinite diameter; Chebyshev constants; pluripotential Green functions; capacities; Vandermondian; Wronskian},
language = {eng},
number = {1},
pages = {293-313},
title = {Transfinite diameter, Chebyshev constants, and capacities in ℂⁿ},
url = {http://eudml.org/doc/280619},
volume = {106},
year = {2012},
}

TY - JOUR
AU - Vyacheslav Zakharyuta
TI - Transfinite diameter, Chebyshev constants, and capacities in ℂⁿ
JO - Annales Polonici Mathematici
PY - 2012
VL - 106
IS - 1
SP - 293
EP - 313
AB - The famous result of geometric complex analysis, due to Fekete and Szegö, states that the transfinite diameter d(K), characterizing the asymptotic size of K, the Chebyshev constant τ(K), characterizing the minimal uniform deviation of a monic polynomial on K, and the capacity c(K), describing the asymptotic behavior of the Green function $g_{K}(z)$ at infinity, coincide. In this paper we give a survey of results on multidimensional notions of transfinite diameter, Chebyshev constants and capacities, related to these classical results and initiated by Leja’s definition of transfinite diameter of a compact set K⊂ ℂⁿ and the author’s paper [Mat. Sb. 25 (1975)], where a multidimensional analog of the Fekete equality d(K) = τ(K) was first considered for any compact set in ℂⁿ. Using some general approach, we introduce an alternative definition of transfinite diameter and show its coincidence with Fekete-Leja’s transfinite diameter. In conclusion we discuss an application of the results of the author’s paper mentioned above to the asymptotics of the leading coefficients of orthogonal polynomial bases in Hilbert spaces related to a given pluriregular polynomially convex compact set in ℂⁿ.
LA - eng
KW - transfinite diameter; Chebyshev constants; pluripotential Green functions; capacities; Vandermondian; Wronskian
UR - http://eudml.org/doc/280619
ER -