Article
Ufa Mathematical Journal
Volume 9, Number 3, pp. 109-117
On a Hilbert space of entire functions
Musin I.Kh.
DOI:10.13108/2017-9-3-109
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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.