Karamata's theorem for regularised Cauchy transforms

TL;DR

This paper establishes Abelian and Tauberian theorems connecting the asymptotic behavior of distribution functions of measures with polynomial growth to their regularised Cauchy transforms, advancing understanding in harmonic analysis.

Contribution

It introduces new Abelian and Tauberian theorems for regularised Cauchy transforms of measures with polynomial growth, linking distribution asymptotics to transform asymptotics.

Findings

Proved Abelian theorems for regularised Cauchy transforms.

Established Tauberian theorems relating asymptotics of measures and transforms.

Extended classical results to measures with polynomial growth at infinity.

Abstract

We prove Abelian and Tauberian theorems for regularised 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 regularised Cauchy transform.