On a Hilbert space of entire functions

I. Kh. Musin

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arXiv · 1710.06143

On a Hilbert space of entire functions

I. Kh. Musin

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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.

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I. Kh. Musin. 2017-10-17. On a Hilbert space of entire functions. https://arxiv.org/abs/1710.06143

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