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7 papers
Representation of functions in locally convex subspaces of $A^\infty (D)$ by series of exponentials
K. P. Isaevab,
K. V. Trounova,
R. S. Yulmukhametovab a Bashkir State University,
Zaki Validi str. 32,
450074, Ufa, Russia
b Institute of Mathematics, Ufa Scientific Center, RAS,
Chernyshevsky str. 112,
450008, Ufa, Russia
Abstract:
Let
$D$ be a bounded convex domain in the complex plane,
$\mathcal M_0=(M_n)_{n=1}^\infty $ be a convex sequence of positive numbers satisfying the “non-quasi-analyticity” condition:
$$
\sum_n\frac {M_n}{M_{n+1}}<\infty,
$$
$\mathcal M_k=(M_{n+k})_{n=1}^\infty$,
$k=0,1,2,3,\ldots$ be the sequences obtained from the initial ones by removing first
$k$ terms. For each sequence
$\mathcal M_0=(M_n)_{n=1}^\infty$ we consider the Banach space
$H(\mathcal M_0,D)$ of functions analytic in a bounded convex domain
$D$ with the norm:
$$
\|f\| ^2=\sup_n \frac 1{M_n^2}\sup_{z\in D}|f^{(n)}(z)|^2.
$$
In the work we study locally convex subspaces in the space of analytic functions in
$D$ infinitely differentiable in
$\overline D$ obtained as the inductive limit of the spaces
$H(\mathcal M_k,D)$. We prove that for each convex domain there exists a system of exponentials
$e^{\lambda_nz}$,
$n\in \mathbb{N}$, such that each function in the inductive limit
$f\in \lim {\text ind}\, H(\mathcal M_k,D):=\mathcal H(\mathcal M_0,D)$ is represented as the series over this system of exponentials and the series converges in the topology
of
$\mathcal H(\mathcal M_0,D)$. The main tool for constructing the systems of exponentials is entire functions with a prescribed asymptotic behavior. The characteristic functions
$L$ with more sharp asymptotic estimates allow us to represent analytic functions by means of the series of the exponentials in the spaces with a finer topology. In the work we construct entire functions with gentle asymptotic estimates. In addition, we obtain lower bounds for the derivatives of these functions at zeroes.
Keywords:
analytic functions, entire functions, subharmonic functions, series of exponentials.
UDC:
517.5
MSC: 30B50,
30D20,
30D60 Received: 01.06.2017
Fulltext:
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