A new proof of Poltoratskii's theorem

Vojkan Jakšić*, Yoram Last

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

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 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, F(z)=∫(x-z)-1 f(x) dμ(x) and Fμ(z)=∫(x-z)-1dμ(x).

Original languageAmerican English
Pages (from-to)103-110
Number of pages8
JournalJournal of Functional Analysis
Volume215
Issue number1
DOIs
StatePublished - 1 Oct 2004

Bibliographical note

Funding 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.

Keywords

  • Borel transform
  • Poltoratskii theorem
  • Rank one perturbations
  • Singular measure
  • Stieltjes transform

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