Karamata's theorem for regularized Cauchy transforms

Matthias Langer, Harald Woracek

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Abstract

We prove Abelian and Tauberian theorems for regularized Cauchy transforms of positive Borel measures on the real line whose distribution functions grow at most polynomially at infinity. In particular, we relate the asymptotics of the distribution functions to the asymptotics of the regularized Cauchy transform.

Original languageEnglish
Pages (from-to)1-61
Number of pages61
JournalProceedings of the Royal Society of Edinburgh: Section A Mathematics
Early online date26 Jan 2024
DOIs
Publication statusE-pub ahead of print - 26 Jan 2024

Keywords

  • regularised Cauchy transform
  • Tauberian theorem
  • Abelian theorem
  • regularly varying function
  • Grommer-Hamburger theorem

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