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On the absolutely continuous component of a weak limit of measures on $ \mathbb{R}$ supported on discrete sets


Author: Alexander Y. Gordon
Journal: Proc. Amer. Math. Soc. 144 (2016), 4743-4752
MSC (2010): Primary 28A33, 44A15
Published electronically: July 21, 2016
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Abstract: Let $ \mu _1,\mu _2,\ldots $ 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.


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Additional Information

Alexander Y. Gordon
Affiliation: Department of Mathematics and Statistics, University of North Carolina at Charlotte, 9201 University City Blvd., Charlotte, North Carolina 28223
Email: aygordon@uncc.edu

DOI: https://doi.org/10.1090/proc/13032
Keywords: Weak convergence of measures, Hilbert transform of a measure, absolutely continuous component of a measure, essential support of an absolutely continuous measure
Received by editor(s): October 28, 2015
Published electronically: July 21, 2016
Communicated by: Michael Hitrik
Article copyright: © Copyright 2016 American Mathematical Society