Two counterexamples involving inner functions

Author:
Kenneth Stephenson

Journal:
Proc. Amer. Math. Soc. **73** (1979), 22-24

MSC:
Primary 30D50

MathSciNet review:
512050

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Abstract: Two questions about the existence of bounded analytic functions with prescribed behavior are answered using a construction technique of McLaughlin and Piranian. If *A* is a relatively closed, countable subset of the unit disc, the construction gives

(I) Function so that the inner factor of is a finite Blaschke product if and only if .

(II) Inner function so that has a discrete singular inner factor if and only if .

**[1]**Carl C. Cowen,*The commutant of an analytic Toeplitz operator*, Trans. Amer. Math. Soc.**239**(1978), 1–31. MR**0482347**, 10.1090/S0002-9947-1978-0482347-9**[2]**Renate McLaughlin and George Piranian,*The exceptional set of an inner function*, Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II**185**(1976), no. 1-3, 51–54. MR**0447585****[3]**Kenneth Stephenson,*Omitted values of singular inner functions*, Michigan Math. J.**25**(1978), no. 1, 91–100. MR**0481015****[4]**James Thomson,*The commutant of a class of analytic Toeplitz operators. II*, Indiana Univ. Math. J.**25**(1976), no. 8, 793–800. MR**0417843**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0512050-4

Keywords:
Bounded analytic function,
inner function,
Riemann surface

Article copyright:
© Copyright 1979
American Mathematical Society