Exponential series in normed spaces of analytic functions
R. A. Bashmakova,
K. P. Isaevb,
A. A. Makhotaa a Bashkir State University, Zaki Validi str. 32, 450074, Ufa, Russia
b Institute of Mathematics, Ufa Federal Research Center, RAS, Chernyshebvsky str. 112, 450008, Ufa, Russia
Abstract:
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
$s=4$.
Keywords:
analytic function, entire function, Fourier–Laplace transform, interpolation, exponential series.
UDC:
517.537+
517.547
MSC: 30B50,
30D20 Received: 08.06.2021
Fulltext:
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