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arXiv · 2110.07558

On a property of Herglotz functions

Nurulla Azamov
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Abstract

In this note I prove the following property of Herglotz functions, which to my knowledge is new: For a Herglotz function $h(z)$ and a real number $r \in \mathbb R$ define a Herglotz function $g_r(z) = (r - h(z))^{-1}.$ Let $\mu_r^{(s)}$ be the singular part of the measure $\mu_r$ which corresponds to $g_r(z)$ via the Herglotz representation theorem. Then the measure $\int_0^1 \mu_r^{(s)}\,dr$ is absolutely continuous, its density is integer-valued a.e., and moreover the density takes values $0$ or $1$ a.e.

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Nurulla Azamov. 2021-10-09. On a property of Herglotz functions. https://arxiv.org/abs/2110.07558

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