Geodesic
On a Hilbert space of entire functions
Ufa mathematical journal, Tome 9 (2017) no. 3, pp. 109-117 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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},
     pages = {109--117},
     year = {2017},
     volume = {9},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a10/}
}
TY  - JOUR
AU  - I. Kh. Musin
TI  - On a Hilbert space of entire functions
JO  - Ufa mathematical journal
PY  - 2017
SP  - 109
EP  - 117
VL  - 9
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a10/
LA  - en
ID  - UFA_2017_9_3_a10
ER  - 
%0 Journal Article
%A I. Kh. Musin
%T On a Hilbert space of entire functions
%J Ufa mathematical journal
%D 2017
%P 109-117
%V 9
%N 3
%U http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a10/
%G en
%F UFA_2017_9_3_a10
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/

[1] I. Kh. Musin, “On a space of entire functions rapidly decreasing on $R^n$ and its Fourier transform”, Concrete Operators, 2:1 (2015), 120–138 | DOI | MR | Zbl

[2] D. Azagra, “Global and fine approximation of convex functions”, Proc. London Math. Soc., 107:4 (2013), 799–824 | DOI | MR | Zbl

[3] D. Azagra, Global approximation of convex functions, arXiv: 1112.1042 [math.FA]

[4] B.A. Taylor, “On weighted polynomial approximation of entire functions”, Pacific Journal of Mathematics, 1971:36 (2), 523–539 | MR

[5] V. V. Napalkov, S. V. Popënov, “On the Laplace transform of functionals in the Bergman weight space of entire functions in $\mathbb{C}^n$”, Dokl. Math., 55:1 (1997), 110–112 | MR | Zbl

[6] S. V. Popenov, “On Laplace transform of functionals in some weighted Bergman spaces in ${\mathbb{C}}^n$”, Proc. Int. Conf. “Complex analysis, differential equations, numerical methods and applications”, v. 2, Ufa, 1996, 125–132 (in Russian)

[7] R. S. Yulmukhametov, “Asymptotics of multi-dimensional Laplace integral”, Studies in approximation theory, Ufa, 1989, 132–137 (in Russian) | MR | Zbl

[8] V. V. Napalkov, R. A. Bashmakov, R. S. Yulmukhametov, “Asymptotic behavior of Laplace integrals and geometric characteristics of convex functions”, Dokl. Math., 75:2 (2007), 190–192 | DOI | MR | Zbl

[9] R. A. Bashmakov, K. P. Isaev, R. S. Yulmukhametov, “On geometric characteristics of convex function and Laplace integrals”, Ufimskij Matem. Zhurn., 2:1 (2010), 3–16 (in Russian) | Zbl

[10] G. M. Fikhtengolts, Course of differential and integral calculus, v. II, Nauka, M., 1970 (in Russian)