On a Hilbert space of entire functions

Abstract

We consider the Hilbert space $F^2_{\varphi}$ of entire functions of $n$ variables constructed by means of a convex function $\varphi$ in $\mathbb{C}^n$ depending on the absolute value of the variable and growing at infinity faster than $a|z|$ for each $a > 0$. We study the problem on describing the dual space in terms of the Laplace transform of the functionals. Under certain conditions for the weight function $\varphi$, we obtain the description of the Laplace transform of linear continuous functionals on $F^2_{\varphi}$. The proof of the main result is based on using new properties of Young-Fenchel transform and one result on the asymptotics of the multi-dimensional Laplace integral established by R. A. Bashmakov, K. P. Isaev, R. S. Yulmukhametov.