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Multiplicity of boundary values of inner functions. (English. Russian original) Zbl 0648.30002
Sov. J. Contemp. Math. Anal. Arm. Acad. Sci. 22, No. 5, 74-87 (1987); translation from Izv. Akad. Nauk Arm. SSR, Mat. 22, No. 5, 490-503 (1987).
The author considers, generally, inner functions which are not finite Blaschke products. For such a function I the multiplicity of a value \(\alpha\) with \(| \alpha | =1\) is defined in terms of the singular part of the Borel measure that defines \((\alpha -I(z))/(\alpha +I(z))\). It is shown that for each I the value \(\alpha\) is assumed either a countable or a continual number of times. Amongst other developments it is shown that the Blaschke product with zeros at \(\{a_ n\}\) has countable multiplicity if \(\sum_{n}(1-| a_ n|)^{1/2}\) converges. Further, if F is a closed subset of the unit circle, and \(\{r_ n\}\) is a positive Blaschke sequence with \(\sum_{n}(1-r_ n)^{1/2}=\infty,\) there is a real sequence \(\{\alpha_ n\}\) such that the Blaschke product with zeros at \(\{r_ ne^{i\alpha_ n}\}\) has F as a multiplicity set for its zeros on the unit circle.
Reviewer: C.N.Linden

30B30 Boundary behavior of power series in one complex variable; over-convergence
30D50 Blaschke products, etc. (MSC2000)
30D55 \(H^p\)-classes (MSC2000)