A reproducing kernel approach to Lebesgue decomposition

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”...

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Bibliographic Details
Published inCanadian journal of mathematics Vol. 77; no. 5; pp. 1570 - 1610
Main Authors Bal, Jashan, Martin, Robert T.W., Naderi, Fouad
Format Journal Article
LanguageEnglish
Published Canada Canadian Mathematical Society 01.10.2025
Subjects
28A99
15A63
Hilbert spaces of analytic functions
sesquilinear forms in Hilbert space
47B32
Reproducing kernels
Lebesgue decomposition
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ISSN0008-414X
1496-4279
DOI10.4153/S0008414X24000488

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Summary: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.
ISSN:0008-414X
1496-4279
DOI:10.4153/S0008414X24000488