Wyniki wyszukiwania

1

Internal characteristics of domains in ℂⁿ

100%

Zakharyuta V.

Annales Polonici Mathematici

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2014

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tom 111

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nr 3

215-236

EN

This paper is devoted to internal capacity characteristics of a domain D ⊂ ℂⁿ, relative to a point a ∈ D, which have their origin in the notion of the conformal radius of a simply connected plane domain relative to a point. Our main goal is to study the internal Chebyshev constants and transfinite diameters for a domain D ⊂ ℂⁿ and its boundary ∂D relative to a point a ∈ D in the spirit of the author's article [Math. USSR-Sb. 25 (1975), 350-364], where similar characteristics have been investigated for compact sets in ℂⁿ. The central notion of directional Chebyshev constants is based on the asymptotic behavior of extremal monic "polynomials" and "copolynomials" in directions determined by the arithmetic of the index set ℤⁿ. Some results are closely related to results on the sth Reiffen pseudometrics and internal directional analytic capacities of higher order (Jarnicki-Pflug, Nivoche) describing the asymptotic behavior of extremal "copolynomials" in varied directions when approaching the point a.

2

Transfinite diameter, Chebyshev constants, and capacities in ℂⁿ

100%

Zakharyuta V.

Annales Polonici Mathematici

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2012

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tom 106

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nr 1

293-313

EN

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.

3

On Pólya's Theorem in several complex variables

63%

Günyüz O., Zakharyuta V.

Banach Center Publications

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2015

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tom 107

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nr 1

149-157

EN

Let K be a compact set in ℂ, f a function analytic in ℂ̅∖K vanishing at ∞. Let $f(z) = ∑_{k=0}^{∞} a_{k}z^{-k-1}$ be its Taylor expansion at ∞, and $H_{s}(f) = det(a_{k+l})_{k,l=0}^{s}$ the sequence of Hankel determinants. The classical Pólya inequality says that $lim sup_{s→∞} |H_{s}(f)|^{1/s²} ≤ d(K)$, where d(K) is the transfinite diameter of K. Goluzin has shown that for some class of compacta this inequality is sharp. We provide here a sharpness result for the multivariate analog of Pólya's inequality, considered by the second author in Math. USSR Sbornik 25 (1975), 350-364.

4

Isomorphic classification of the tensor products $E₀(exp αi) ⊗̂ E_{∞}(exp βj)$

63%

Chalov P., Zakharyuta V.

Studia Mathematica

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2011

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tom 204

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nr 3

275-282

EN

It is proved, using so-called multirectangular invariants, that the condition αβ = α̃β̃ is sufficient for the isomorphism of the spaces $E₀(exp αi) ⊗̂ E_{∞}(exp βj)$ and $E₀(exp α̃i) ⊗̂ E_{∞}(exp β̃j)$. This solves a problem posed in [14, 15, 1]. Notice that the necessity has been proved earlier in [14].

Ograniczanie wyników

2 Annales Polonici Mathematici

1 Banach Center Publications

1 Studia Mathematica

4 Zakharyuta V.

1 Chalov P.

1 Günyüz O.

1 2015

1 2014

1 2012

1 2011

Internal characteristics of domains in ℂⁿ

Transfinite diameter, Chebyshev constants, and capacities in ℂⁿ

On Pólya's Theorem in several complex variables

Isomorphic classification of the tensor products $E₀(exp αi) ⊗̂ E_{∞}(exp βj)$