Le Hai Khoi

Dual of the function algebra

In this paper we present the following results: a description, via the Laplace transformation of analytic functionals, of the dual to the (DFS)-space ( being either a bounded -smooth convex domain in , with , or a bounded convex domain in ) as an (FS)-space of entire functions satisfying a certain growth condition; an explicit construction of a countable sufficient set for ; and a possibility of representating functions from in the form of Dirichlet series.

A^{-\infty }(D)

In the space

and representation of functions in Dirichlet series

A^{-\infty} (\Bbb D)

Weakly sufficient sets for A−∞(D)

of functions of polynomial growth, weakly sufficient sets are those such that the topology induced by restriction to the set coincides with the topology of the original space. Horowitz, Korenblum and Pinchuk defined sampling sets for

On an explicit construction of weakly sufficient sets for the function algebra A −∞(Ω)

A^{-\infty} (\Bbb D)

On the duality between and for convex domains

as those such that the restriction of a function to the set determines the type of growth of the function. We show that sampling sets are always weakly sufficient, that weakly sufficient sets are always of uniqueness, and provide examples of discrete sets that show that the converse implications do not hold

COMPACT DIFFERENCES OF COMPOSITION OPERATORS ON BERGMAN SPACES IN THE BALL

This article studies the space A −∞(Ω), the function algebra of holomorphic functions on a domain Ω with 𝒞1 boundary in ℂ n with polynomial growth. In particular, we present an explicit construction of countable subset that is weakly sufficient.

Minimal absolutely representing systems of exponentials for A−∞(Ω)

The goal of this Note is to prove that the Laplace transformation of analytic functionals establishes the mutual duality between the spaces A−∞(D) and AD−∞ (D being a bounded convex domain in CN) and that functions from AD−∞ can be represented in a form of Dirichlet series with frequencies from D. To cite this article: A.V. Abanin, L.H. Khoi, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Mutual dualities between A −∞(Ω) and for lineally convex domains

We obtain necessary and sufficient conditions for the compactness of differences of composition operators acting on the weighted Bergman spaces in the unit ball. A representation of a composition operator as a finite sum of composition operators modulo compact operators is also studied. 2010 Mathematics subject classification: primary 32A36; secondary 47B33.

On the Structure of \(\mathcal {N}_{p}\)-Spaces in the Ball

Abstract We study a minimal property of absolutely representing systems of exponentials E Λ = ( e λ k z ) in the space A − ∞ ( Ω ) of holomorphic functions in a bounded convex domain Ω of the complex plane C , with polynomial growth near the boundary. A relationship between the absolutely representing property and absolutely nontrivial expansions of zero in A − ∞ ( Ω ) is also investigated.