Zikkos, Elias Closure of the linear span of an exponential system in a weighted Banach space. (English) Zbl 1424.30011 J. Class. Anal. 10, No. 2, 131-146 (2017). 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 \[ \sum_{n=1}^\infty\left(\sum^{\mu_n-1}_{k=0}c_{n,k}z^k\right)e^{\lambda_nz}, \quad c_{n,k}\in \mathbb C, \quad\forall z \in \mathbb C. \] Cited in 1 Document 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 Keywords:completeness; closure; minimality; Taylor-Dirichlet series PDFBibTeX XMLCite \textit{E. Zikkos}, J. Class. Anal. 10, No. 2, 131--146 (2017; Zbl 1424.30011) Full Text: DOI References: [1] J. M. ANDERSON, K. G. BINMORE, {\it Closure theorems with applications to entire functions with gaps}, Trans. Amer. Math. Soc. 161 (1971) 381-400. · Zbl 0224.30007 [2] G. T. DENG, {\it Incompleteness and closure of a linear span of exponential system in a weighted Banach} {\it space}, J. Approx. Theory 125 no. 1 (2003), 1-9. · Zbl 1036.30002 [3] W. H. J. FUCHS, {\it On the closure of}{\it{e}{\it −t}{\it t}{\it a}{\it n}{\it }}, Proc. Cambridge Philos. Soc. 42 (1946), 91-105. · Zbl 0061.13401 [4] P. MALLIAVIN, {\it Sur quelques proc´ed´es d’extrapolation}, Acta Math. 93 (1955) 179-255. · Zbl 0067.05104 [5] J. NING, G. T. DENG, C. YI, {\it Incompleteness and closure of the multiplicity system}{\it{t}{\it k}{\it e}λ{\it j}{\it t}{\it } in the} {\it weighted Banach space}, J. Math. Anal. Appl. 341 no. 2 (2008), 1007-1017. · Zbl 1162.30005 [6] X. YANG, {\it Incompleteness of exponential system in the weighted Banach space}, J. Approx. Theory 153 no. 1 (2008), 73-79. · Zbl 1149.30025 [7] E. ZIKKOS, {\it Completeness of an exponential system in weighted Banach spaces and closure of its} {\it linear span}, J. Approx. Theory 146 no. 1 (2007), 115-148. · Zbl 1113.30005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.