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Universal series in \({\bigcap}_{p>1}{\ell}^p\). (English) Zbl 1195.30055

Let \(X\) be a topological vector space over the field \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C},\) the topology of which is induced by a translation invariant metric \(\rho.\) Let \(\{x_j\}_{j\in \mathbb{N}}\) be a fixed sequence in \(X\). A sequence \(\{a_j\}_{j\in \mathbb{N}}\subset \mathbb{K}^{\mathbb{N}}\) belongs to the class \(\mathcal{U}\) if the sequence \(\{\sum _{j=0}^{n}a_jx_j \}_{n\in \mathbb{N}}\) is dense in \(X.\) In the paper the following theorem is proved.
Theorem. Let \(E\) be a topological vector space over \(\mathbb{K}\) whose topology is induced by a translation invariant metric \(\rho.\) Let \(\{x_j\}_{j\in \mathbb{N}}\) be a fixed sequence in \(E\) satisfying the following condition: for every finite set \(I\subset \mathbb{N},\) there exist distinct indices \(j_n(i),\) with \(n \in \mathbb{N}\) and \(i \in I,\) such that \(x_{j_n(i)} \to x_i\) as \(n \to \infty.\) Let \(X\) denote the set of all finite linear combinations of the elements of the sequence \(\{x_j\}_{j\in \mathbb{N}}\) endowed with the metric \(\rho.\) Then \(\mathcal{U}\neq \emptyset\) and there exists a scalar sequence \(\mathbf{a}=\{a_j\}_{j\in \mathbb{N}}\) in \(\mathbb{K}^{\mathbb{N}}\) such that the sequence \(\{\sum _{j=0}^{n}a_jx_j \}_{n\in \mathbb{N}}\) is dense in \(X.\) The sequence \(\mathbf{a}\) can be chosen in \(\bigcap_{p>1} l^p\).
As an application of the results, the authors obtain a unification of some known results related to approximation by translates of specific functions including the Riemann \(\zeta\)-function. Another application gives universal trigonometric series in \(\mathbb{R}^{\nu}\) simultaneously with respect to all \(\sigma\)-finite Borel measures in \(\mathbb{R}^{\nu}\). Stronger results are obtained by using Dirichlet series.

MSC:

30E10 Approximation in the complex plane
41A30 Approximation by other special function classes
41A45 Approximation by arbitrary linear expressions
42A10 Trigonometric approximation
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