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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

##### MSC:
 30B30 Boundary behavior of power series in one complex variable; over-convergence 30D50 Blaschke products, etc. (MSC2000) 30D55 $$H^p$$-classes (MSC2000)
##### Keywords:
boundary zeros; inner functions; Blaschke products