On the absolutely continuous component of a weak limit of measures on $\mathbb {R}$ supported on discrete sets
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- by Alexander Y. Gordon
- Proc. Amer. Math. Soc. 144 (2016), 4743-4752
- DOI: https://doi.org/10.1090/proc/13032
- 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.References
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Bibliographic Information
- Alexander Y. Gordon
- Affiliation: Department of Mathematics and Statistics, University of North Carolina at Charlotte, 9201 University City Blvd., Charlotte, North Carolina 28223
- MR Author ID: 239917
- Email: aygordon@uncc.edu
- Received by editor(s): October 28, 2015
- Published electronically: July 21, 2016
- Communicated by: Michael Hitrik
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4743-4752
- MSC (2010): Primary 28A33, 44A15
- DOI: https://doi.org/10.1090/proc/13032
- MathSciNet review: 3544526