A reproducing kernel approach to Lebesgue decomposition

2023/12/04 by Jashan Bal, Robert T. W. Martin, Bal, Jashan +3 · 1 citation
Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.2312.01961

openalex publication_date 2023/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that properties of pairs of finite, positive and regular Borel measures on the complex unit circle such as domination, absolute continuity and singularity can be completely described in terms of containment and intersection of their reproducing kernel Hilbert spaces of `Cauchy transforms' in the complex unit disk. This leads to a new construction of the classical Lebesgue decomposition and proof of the Radon--Nikodym theorem using reproducing kernel theory and functional analysis.

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• Universal Kernel Models for Iterated Completely Positive Maps

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