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On the absolutely continuous component of a weak limit of measures on \(\mathbb R\) supported on discrete sets. (English) Zbl 1351.44003

Summary: Let \(\mu_1\), \(\mu_2,\dots\) be a sequence of positive Borel measures on \(\mathbb R\) each of which is supported on a set having no finite limit points. Suppose the sequence \(\mu_n\) weakly converges to a Borel measure \(\nu\). Let \(\nu_{\mathrm{ac}}\) be the absolutely continuous component of \(\nu\), and \(X\subset \mathbb R\) the essential support of \(\nu_{\mathrm{ac}}\). We characterize the set \(X\) in terms of the limiting behavior of the Hilbert transforms of the measures \(\mu_n\). Potential applications include those in spectral theory.

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
28A33 Spaces of measures, convergence of measures
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References:

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