ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

#### For citation:

Krivosheev A. S., Krivosheeva O. A. Representation of functions on a line by a series of exponential monomials. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 4, pp. 416-429. DOI: 10.18500/1816-9791-2022-22-4-416-429

Published online:
30.11.2022
Full text:
Language:
Russian
Article type:
Article
UDC:
517.98
EDN:
TPUWZW

# Representation of functions on a line by a series of exponential monomials

Autors:
Krivosheev Alexander Sergeevich, Institute of Mathematics with Computing Centre
Krivosheeva Olesya Alexandrovna, Bashkir State University
Abstract:

In this work, we consider the weight spaces of integrable functions $L_p^\omega$ ($p\geq 1$) and continuous functions $C^\omega$ on the real line. Let $\Lambda=\{\lambda_k,n_k\}$ be an unbounded increasing sequence of positive numbers $\lambda_k$ and their multiplicities $n_k$, $\mathcal{E}(\Lambda)=\{t^n e^{\lambda_k t}\}$ be a system of exponential monomials constructed from the sequence $\Lambda$. We study the subspaces $W^p (\Lambda,\omega)$ and $W^0 (\Lambda,\omega)$, which are the closures of the linear span of the system $\mathcal{E}(\Lambda)$ in the spaces $L_p^\omega$ and $C^\omega$, respectively. Under natural constraints on $\Lambda$ (the finiteness of the condensation index $S_\Lambda$ and $n_k/\lambda_k\leq c$, $k\geq 1$) and on the convex weight $\omega$, conditions are obtained under which each function of these subspaces continues to an entire function and is represented by a series in the system $\mathcal{E}(\Lambda)$ that converges absolutely and uniformly on compact sets in the plane. In contrast to the previously known results for the specified representation problem, we do not require that the sequence $\Lambda$ has a density, and we do not impose the separability condition: $\lambda_{k+1}-\lambda_k\geq h$, $k\geq 1$ (instead, the condition of equality to zero of the special condensation index is used).

Key words:
Acknowledgments:
The work of O. A. Krivosheeva is supported in part by the Young Russian Mathematics award.
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