On a Hilbert space of entire functions

TL;DR

This paper studies a weighted Hilbert space of entire functions in several variables, describing its dual space via Laplace transforms and employing advanced convex analysis techniques.

Contribution

It provides a new description of the dual space of a weighted Hilbert space of entire functions using Laplace transforms under certain convexity conditions.

Findings

Characterization of the dual space via Laplace transforms.

Application of Young-Fenchel transformation properties.

Asymptotic analysis of multidimensional Laplace transforms.

Abstract

A weighted Hilbert space $F^2_{\varphi}$ of entire functions of $n$ variables is considered in the paper. The weight function $\varphi$ is a convex function on ${\mathbb C}^n$ depending on modules of variables and growing at infinity faster than $a \Vert z \Vert$ for each $a > 0$. The problem of description of the strong dual of this space in terms of the Laplace transformation of functionals is studied in the article. Under some additional conditions on $\varphi$ the space of the Laplace transforms of linear continuous functionals on $F^2_{\varphi}$ is described. The proof of the main result is based on new properties of the Young-Fenchel transformation and a result of R.A. Bashmakov, K.P. Isaev and R.S. Yulmukhametov on asymptotics of multidimensional Laplace transform.