Geodesic
A reproducing kernel approach to Lebesgue decomposition
Canadian journal of mathematics, Tome 77 (2025) no. 5, pp. 1570-1610

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DOI

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.
DOI : 10.4153/S0008414X24000488
Mots-clés : Reproducing kernels, Hilbert spaces of analytic functions, Lebesgue decomposition, sesquilinear forms in Hilbert space 
Bal, Jashan; Martin, Robert T.W.; Naderi, Fouad. A reproducing kernel approach to Lebesgue decomposition. Canadian journal of mathematics, Tome 77 (2025) no. 5, pp. 1570-1610. doi: 10.4153/S0008414X24000488
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     title = {A reproducing kernel approach to {Lebesgue} decomposition},
     journal = {Canadian journal of mathematics},
     pages = {1570--1610},
     year = {2025},
     volume = {77},
     number = {5},
     doi = {10.4153/S0008414X24000488},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X24000488/}
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