Karamata's theorem for regularized Cauchy transforms

Matthias Langer; Harald Woracek

doi:10.1017/prm.2023.128

Karamata's theorem for regularized Cauchy transforms

Matthias Langer, Harald Woracek

Mathematics And Statistics

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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 language	English
Pages (from-to)	1431-1491
Number of pages	61
Journal	Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Volume	155
Issue number	4
Early online date	26 Jan 2024
DOIs	https://doi.org/10.1017/prm.2023.128
Publication status	Published - 13 Aug 2025

Funding

The second author was supported by the joint project I 4600 of the Austrian Science Fund (FWF) and the Russian foundation of basic research (RFBR).

Keywords

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

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10.1017/prm.2023.128Licence: CC BY 4.0

Langer-Woracek-PRSEA-2024-Karamatas-theorem-for-regularised-Cauchy-transformsFinal published version, 885 KBLicence: CC BY 4.0

Cite this

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title = "Karamata's theorem for regularized Cauchy transforms",

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.",

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AU - Woracek, Harald

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AB - 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.

KW - regularised Cauchy transform

KW - Tauberian theorem

KW - Abelian theorem

KW - regularly varying function

KW - Grommer-Hamburger theorem

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UR - https://www.scopus.com/pages/publications/85183928095

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M3 - Article

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JO - Proceedings of the Royal Society of Edinburgh: Section A Mathematics

JF - Proceedings of the Royal Society of Edinburgh: Section A Mathematics

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ER -

Karamata's theorem for regularized Cauchy transforms

Abstract

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Keywords

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