The Representation by Series of Exponential Monomials of Functions from Weight Subspaces on a Line

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, we use the condition of the finiteness of the condensation index).

Sufficient conditions for the incompleteness of the system $\mathcal{E}(\Lambda)$ in the spaces $L_{p}^{\omega}$ and $C^{\omega}$ are also obtained.