03-353 Vojkan Jaksic and Yoram Last

A New Proof of Poltoratskii's Theorem (127K, Postscript) Jul 30, 03

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Abstract. We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure $\mu$ on $\R$ and any $f\in L^1(\R,d\mu)$, $\lim_{\epsilon\to 0}(F_{f\mu}(E+i\epsilon)/F_{\mu}(E+i\epsilon)) = f(E)$ a.e. w.r.t. $\musing$, where $\musing$ is the part of $\mu$ which is singular with respect to Lebesgue measure and $F$ denotes a Borel transform, namely, $F_{f\mu}(z) = \int (x-z)^{-1}f(x)\,d\mu(x)$ and $F_{\mu}(z) = \int (x-z)^{-1}d\mu(x)$.

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