Geodesic
Distribution of Zeros of Exponential-Type Entire Functions with Constraints on Growth along a Line
Matematičeskie zametki, Tome 108 (2020) no. 4, pp. 588-600.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $g\ne 0$ be an entire function of exponential type in the complex plane $\mathbb C$, and let ${\mathsf Z}=\{{\mathsf z}_k\}_{k=1,2,\dots}$ be a sequence of points in $\mathbb C$. We give a criterion for the existence of an entire function $f\ne 0$ of exponential type which vanishes on ${\mathsf Z}$ and satisfies the constraint $$ \ln |f(iy)|\le \ln |g(iy)|+o(|y|),\qquad y\to \pm\infty. $$ Our results generalize and develop joint results of P. Malliavin and L. A. Rubel. Applications to multipliers for entire functions of exponential type, to analytic functionals and their convolutions in the complex plane, and to the completeness problem for exponential systems in spaces of locally analytic functions on compact spaces in terms of the widths of these spaces are given.
Keywords: zeros of entire function, analytic functional, completeness of exponential systems.
Mots-clés : multiplier, convolution 
@article{MZM_2020_108_4_a9,
     author = {A. E. Salimova and B. N. Khabibullin},
     title = {Distribution of {Zeros} of {Exponential-Type} {Entire} {Functions} with {Constraints} on {Growth} along a {Line}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {588--600},
     publisher = {mathdoc},
     volume = {108},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_108_4_a9/}
}
TY  - JOUR
AU  - A. E. Salimova
AU  - B. N. Khabibullin
TI  - Distribution of Zeros of Exponential-Type Entire Functions with Constraints on Growth along a Line
JO  - Matematičeskie zametki
PY  - 2020
SP  - 588
EP  - 600
VL  - 108
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2020_108_4_a9/
LA  - ru
ID  - MZM_2020_108_4_a9
ER  - 
%0 Journal Article
%A A. E. Salimova
%A B. N. Khabibullin
%T Distribution of Zeros of Exponential-Type Entire Functions with Constraints on Growth along a Line
%J Matematičeskie zametki
%D 2020
%P 588-600
%V 108
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2020_108_4_a9/
%G ru
%F MZM_2020_108_4_a9
A. E. Salimova; B. N. Khabibullin. Distribution of Zeros of Exponential-Type Entire Functions with Constraints on Growth along a Line. Matematičeskie zametki, Tome 108 (2020) no. 4, pp. 588-600. http://geodesic.mathdoc.fr/item/MZM_2020_108_4_a9/

[1] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, Gostekhizdat, M., 1956 | MR | Zbl

[2] B. Ya. Levin, Lectures on Entire Functions, Transl. Math. Monogr., 150, Amer. Math. Soc., Providence RI, 1996 | MR | Zbl

[3] P. Malliavin, L. A. Rubel, “On small entire functions of exponential type with given zeros”, Bull. Soc. Math. France, 89 (1961), 175–206 | DOI | MR | Zbl

[4] B. N. Khabibullin, “O roste vdol pryamoi tselykh funktsii eksponentsialnogo tipa s zadannymi nulyami”, Analysis Math., 17:3 (1991), 239–256 | DOI | Zbl

[5] B. N. Khabibullin, Raspredelenie nulei tselykh funktsii i vymetanie, Dis. $\dots$ dokt. fiz.-matem. nauk, FTINT, Kharkov, 1993; ПОМИ РАН, Санкт-Петербург, 1994

[6] B. N. Khabibullin, Polnota sistem eksponent i mnozhestva edinstvennosti, RITs BashGU, Ufa, 2012 | DOI

[7] L. A. Rubel, Entire and Meromorphic Functions, Springer, New York, 1996 | MR | Zbl

[8] B. N. Khabibullin, “O roste tselykh funktsii eksponentsialnogo tipa vdol mnimoi osi”, Dokl. AN SSSR, 302:2 (1988), 270–273 | MR | Zbl

[9] B. N. Khabibullin, “O roste tselykh funktsii eksponentsialnogo tipa vdol mnimoi osi”, Matem. sb., 180:5 (1989), 706–719 | MR | Zbl

[10] L. Hörmander, Notions of Convexity, Progr. Math., 127, Birkhäuser Boston, Boston, MA, 1994 | MR | Zbl

[11] V. V. Napalkov, Uravneniya svertki v mnogomernykh prostranstvakh, Nauka, M., 1982 | MR | Zbl

[12] Zh. Sebashtyan-i-Silva, “O nekotorykh klassakh lokalno vypuklykh prostranstv, vazhnykh v prilozheniyakh”, Matematika, 1:1 (1957), 60–77 | MR | Zbl

[13] L. V. Kantopovich, G. P. Akilov, Funktsionalnyi analiz v nopmipovannykh ppostpanstvakh, Fizmatgiz, M., 1959 | MR | Zbl

[14] B. N. Khabibullin, A. V. Shmeleva, “Vymetanie mer i subgarmonicheskikh funktsii na sistemu luchei. I. Klassicheskii sluchai”, Algebra i analiz, 31:1 (2019), 156–210