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 μ on ℝ and any f∈L1(ℝ,dμ),limε→0 (Ffu(E+iε)/Fμ(E+iε))=f(E) a.e. w.r.t. μsing, where μsing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, Ffμ(z)=∫(x-z)-1 f(x) dμ(x) and Fμ(z)=∫(x-z)-1dμ(x).
Bibliographical noteFunding Information:
V.J.’s work was partially supported by NSERC. Y.L.’s work was partially supported by The Israel Science Foundation (Grant 188/02). We thank Barry Simon for motivating this work and for useful discussions.
- Borel transform
- Poltoratskii theorem
- Rank one perturbations
- Singular measure
- Stieltjes transform