Exponential series in normed spaces of analytic functions

R. A. Bashmakov   ; K. P. Isaev   ; A. A. Makhota

Ufa mathematical journal, Tome 13 (2021) no. 3, pp. 27-35

Résumé

There is a classical well-known theorem by A.F. Leontiev on representing functions analytic in a convex domain $D$ and continuous up to the boundary by series of form $\sum_{k=1}^\infty f_ke^{\lambda_kz}$ converging in the topology of the space $H(D)$, that is, uniformly on compact subsets in $D$. In the paper we prove the possibility of representing the functions in \begin{equation*} A_0(D)=\left \{f\in H(D)\bigcap C(\overline D):\ \|f \|:=\sup_{z\in \overline D}|f(z)|\right \} \end{equation*} by the exponential series converging in a stronger topology, namely, there exists an integer number $s>0$ such that 1) for each bounded convex domain $D$ there exists a system of exponentials $e^{\lambda_kz},$ ${k\in \mathbb{N}}$, such that each function $f\in H(D)\bigcap C^{(s)}(\overline D)$ is represented as a series over this system converging in the norm of the space $A_0(D)$; 2) for each bounded convex domain $D$ there exists a system of exponentials $e^{\lambda_kz},$ ${ k\in \mathbb{N}}$ such that each function $f\in A_0(D)$ is represented as a series over this system converging in the norm \begin{equation*} \|f\| = \sup_{z\in D}|f(z)|(d(z))^s, \end{equation*} where $d(z)$ is the distance from a point $z$ to the boundary of the domain $D$. The number $s$ is related with the existence of entire functions with a maximal possible asymptotic estimate. In particular cases, when $D$ is a polygon or a domina with a smooth boundary possessing a smooth curvature separated from zero, we can assume that $s=4$.

Détail

Keywords: analytic function, entire function, exponential series.
Mots-clés : Fourier–Laplace transform, interpolation

Bibliographie

[1] A. F. Leontiev, Exponential series, Nauka, M., 1976 (in Russian)

[2] D. L. Russell, “On exponential bases for the Sobolev spaces over an interval”, J. Math. Anal. Appl., 87:2 (1982), 528–550 | DOI | Zbl

[3] B. Ya. Levin, Yu. I. Lyubarskii, “Interpolation by means of special classes of entire functions and related expansions in series of exponentials”, Math. USSR-Izv., 9:3 (1975), 621–662 | DOI | Zbl

[4] K. P. Isaev, “Riesz bases of exponents in Bergman spaces on convex polygons”, Ufimskij Matem. Zhurn., 2:1 (2010), 71–86 (in Russian) | Zbl

[5] V. I. Lutsenko, Unconditional bases of exponentials in Smirnov spaces, PhD thesis, Institute of Mathematics, Ufa Scientific Center, RAS, 1992 (in Russian)

[6] K. P. Isaev, R. S. Yulmukhametov, “The absence of unconditional bases of exponentials in Bergman spaces on non-polygonal domains”, Izv. Math., 71:6 (2007), 1145–1166 | DOI | Zbl

[7] K. P. Isaev, “On entire functions with given asymptotic behavior”, Probl. Anal. Issues Anal., 7(25):2 (2018), 12–30 | DOI | Zbl

[8] Yu. I. Lyubarskii, “Exponential series in Smirnov spaces and interpolation by entire functions of special classes”, Izv. AN SSSR. Ser. Matem., 52:3 (1988), 559–580 | DOI