Geodesic
Representation of functions on a line by a series of exponential monomials
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 22 (2022) no. 4, pp. 416-429.

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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). 
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A. S. Krivosheev; O. A. Krivosheeva. Representation of functions on a line by a series of exponential monomials. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 22 (2022) no. 4, pp. 416-429. http://geodesic.mathdoc.fr/item/ISU_2022_22_4_a0/

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