Measures whose supports do not have periodic holes (Q2782559)

It was shown in [\textit{H. J. Landau}, Bull. Am. Math. Soc. 70, 566-569 (1964; Zbl 0131.06401)] that there exists a small perturbation \(\{\lambda_n\}\) of the set of integers \(\mathbb{Z}\) such that the system \(\{e^{i\lambda_ nx}\}\) is complete in \(C(U)\) for every (however large) bounded open set \(U\subset\mathbb{R}\) whose closure is disjoint from \(2\pi\mathbb{Z}\). In the present paper, this phenomenon is related to a certain quasi-analyticity for measures on \(\mathbb{R}\). Let \(M_1\) denote the collection of all nontrivial complex-valued measures with bounded support on \(\mathbb{R}\), \(\widehat\mu\) be the Fourier-Laplace transform of \(\mu\in M_1\), and NEWLINE\[NEWLINEE(Z_{\widehat\mu})= \{x^j e^{i\lambda x}:\widehat\mu(\lambda)= 0, 0\leq j\leq\mu(\lambda)\},NEWLINE\]NEWLINE \(\mu(\lambda)\) being the multiplicity of zero at \(\lambda\). It is proved that, given a closed subset \(S\) of \(\mathbb{R}\), the system \(E(Z_{\widehat\mu})\) is complete in \(L^2(U)\) for every bounded open set \(U\) such that \(\overline U\cap S=\emptyset\), if and only if \(\text{supp}(\mu*\nu)\cap S\neq\emptyset\) for each \(\nu\in M_1\). This leads to the construction of various exponential systems of such a type with respect to discrete periodic sets \(S\).NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].