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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 $ f \in {H^\infty }$ so that the inner factor of $ f - \alpha $ is a finite Blaschke product if and only if $ \alpha \in A$.

(II) Inner function $ \phi $ so that $ \phi - \alpha $ has a discrete singular inner factor if and only if $ \alpha \in A$.

References [Enhancements On Off] (What's this?)

  • [1] Carl. C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978), 1-31. MR 0482347 (58:2420)
  • [2] Renate McLaughlin and George Piranian, The exceptional set of an inner function, Sitzungsber. Österreich. Akad. Wiss. Math.-Natur. Kl., Abteilung II, Band 185, 1976. MR 0447585 (56:5895)
  • [3] Kenneth Stephenson, Omitted values of singular inner functions, Michigan Math. J. 25 (1978), 91-100. MR 0481015 (58:1162)
  • [4] James E. Thomson, The commutant of a class of analytic Toeplitz operators. II, Indiana Univ. Math. J. 25 (1976), 793-800. MR 0417843 (54:5891)

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Keywords: Bounded analytic function, inner function, Riemann surface
Article copyright: © Copyright 1979 American Mathematical Society

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