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Value distribution and spectral theory. (English) Zbl 0809.34037

Summary: We develop the theory of value distribution for the class of functions defined by boundary values of Herglotz functions. Examples include boundary values of resolvents of Hilbert space operators, and boundary values of the \(m\)-functions for differential operators. Precise expressions are derived for value distribution, and related to spectral properties of the corresponding measures. As special cases, we evaluate \(| F^{-1}_ 0(S)|\) where the corresponding measure \(\mu\) is purely singular or finite, and construct the local probability density for value distribution where \(\mu\) has an absolutely continuous component.

MSC:

34B20 Weyl theory and its generalizations for ordinary differential equations
30D40 Cluster sets, prime ends, boundary behavior
47A11 Local spectral properties of linear operators
34M99 Ordinary differential equations in the complex domain
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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