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Extendible bases and Kolmogorov problem on asymptotics of entropy and widths of some class of analytic functions

Vyacheslav Zakharyuta — 2011

Annales de la faculté des sciences de Toulouse Mathématiques

Let K be a compact set in an open set D on a Stein manifold Ω of dimension n . We denote by H D the Banach space of all bounded and analytic in D functions endowed with the uniform norm and by A K D a compact subset of the space C K consisted of all restrictions of functions from the unit ball 𝔹 H D . In 1950ies Kolmogorov posed a problem: does ε A K D τ ln 1 ε n + 1 , ε 0 , where ε A K D is the ε -entropy of the compact A K D . We give here a survey of results concerned with this problem and a related problem on the strict asymptotics...

Internal characteristics of domains in ℂⁿ

Vyacheslav Zakharyuta — 2014

Annales Polonici Mathematici

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...

Transfinite diameter, Chebyshev constants, and capacities in ℂⁿ

Vyacheslav Zakharyuta — 2012

Annales Polonici Mathematici

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...

Isomorphic classification of the tensor products E ( e x p α i ) ̂ E ( e x p β j )

Peter ChalovVyacheslav Zakharyuta — 2011

Studia Mathematica

It is proved, using so-called multirectangular invariants, that the condition αβ = α̃β̃ is sufficient for the isomorphism of the spaces E ( e x p α i ) ̂ E ( e x p β j ) and E ( e x p α ̃ i ) ̂ E ( e x p β ̃ j ) . This solves a problem posed in [14, 15, 1]. Notice that the necessity has been proved earlier in [14].

On Pólya's Theorem in several complex variables

Ozan GünyüzVyacheslav Zakharyuta — 2015

Banach Center Publications

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 ) = d e t ( a k + l ) k , l = 0 s the sequence of Hankel determinants. The classical Pólya inequality says that l i m s u p s | H s ( f ) | 1 / s ² 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.

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