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Closure of the linear span of an exponential system in a weighted Banach space. (English) Zbl 1424.30011

Summary: For a certain class of sequences with repeated terms, \[ \{\lambda_n,\mu_n\}^\infty_{n=1} := \{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 \mathrm{ times}},\underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 \mathrm{ times}},\dots,\underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k \mathrm{ times}},\dots\}, \] we prove that every function belonging to the closed span of the exponential system \[ \{x^ke^{\lambda_n x}: n\in \mathbb N,\quad k = 0,1,2,\dots,\mu_n-1\}, \] in some weighted Banach spaces on the real line, extends analytically as an entire function by admitting a series representation of the form \[ \sum_{n=1}^\infty\left(\sum^{\mu_n-1}_{k=0}c_{n,k}z^k\right)e^{\lambda_nz}, \quad c_{n,k}\in \mathbb C, \quad\forall z \in \mathbb C. \]

MSC:

30B50 Dirichlet series, exponential series and other series in one complex variable
30B60 Completeness problems, closure of a system of functions of one complex variable
46E15 Banach spaces of continuous, differentiable or analytic functions
46E20 Hilbert spaces of continuous, differentiable or analytic functions
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References:

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