Geodesic
On a Hilbert space of entire functions
Ufa mathematical journal, Tome 9 (2017) no. 3, pp. 109-117

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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.
Keywords: Hilbert space, entire functions, convex functions, Young–Fenchel transform.
Mots-clés : Laplace transform 
@article{UFA_2017_9_3_a10,
     author = {I. Kh. Musin},
     title = {On a {Hilbert} space of entire functions},
     journal = {Ufa mathematical journal},
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     number = {3},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a10/}
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I. Kh. Musin. On a Hilbert space of entire functions. Ufa mathematical journal, Tome 9 (2017) no. 3, pp. 109-117. http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a10/