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

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.