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The closed span of some exponential system in weighted Banach spaces on the real line and a moment problem. (English) Zbl 1424.30012

This paper is about the convergence and density of the exponential polynomials \[g_m(z)=\sum_{n=1}^m\left(\sum_{k=0}^{\mu_n-1} c_{n,k}z^k\right)e^{\lambda_n z},~~c_{n,k}\in\mathbb{C},\]characterized by the sequence \(\Lambda=(\lambda_k,\mu_k)\) with \(\lambda_k\) positive and strictly increasing.
Let \(E_\Lambda=\{z^ne^{\lambda_kz}\}\) denote the corresponding generating system. Two Banach spaces are considered: \(C_w\) where \(w\) is a positive weight and \(\Vert f\Vert_{C_w}=\sup\{\vert f(t)e^{-w(t)}\vert :t\in\mathbb{R}\}\) and \(L_w^p\) with \(\Vert f\Vert_{L_w^p}=\left(\int_{\mathbb{R}}\vert f(t)e^{-w(t)}\vert ^p\right)^{1/p}\), \(p\ge1\). The main result of the paper concerns conditions prescribed on \(\Lambda\) and \(w\) such that any function in the closed span of \(E_\Lambda\) in any of the two Banach spaces can be analytically extended to an entire function in \(\mathbb{C}\). This holds for example when \(\lambda_k=p_k\) and \(\mu_k=p_{k+1}-p_k\) where \(p_k\) are the successive prime numbers and when \(w\) satisfies some additional conditions. In the special case of the Hilbert space \(L_w^2\), it is proved that a unique system \((r_{n,k})\) exists that is biorthogonal to \(E_\Lambda\). Finally, it is shown that a given sequence \(\{d_{n,k}:n\in\mathbb{N},k=1,\ldots,\mu_k-1\}\), bounded in a sense adapted to \(\Lambda\) can be considered as a moment sequence with respect to the system \(E_\Lambda\) for some function continuous on \(\mathbb{R}\) and vanishing at \(\pm\infty\). Also a special case of an unweighted polynomial moment problem on \([0,\infty)\) is considered when \(E_\Lambda\) is the set of polynomials (with for example moment sequences \(d_n=n^n\) or \(d_n=n!\)).

MSC:

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