Karamata's theorem for regularised Cauchy transforms

2023/07/23 by Langer, Matthias, Woracek, Harald
#26A12 #30E20 #40E05 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.12249

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.

Related

• A continuity theorem for generalised signed measures with an application to Karamata's Tauberian theorem
• A general quantified Ingham-Karamata Tauberian theorem
• Multidimensional Tauberian theorems for wavelet and non-wavelet transforms
• Tauberian theorems for the Stockwell transform of Lizorkin distributions
• Constructions of some families of smooth Cauchy transforms