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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A convergence question in $ H\sp{p}$


Author: Stephen Scheinberg
Journal: Proc. Amer. Math. Soc. 30 (1971), 120-124
MSC: Primary 30.67
MathSciNet review: 0283206
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Abstract: Let $ \phi \in {H^p}$ (unit disc), $ 0 < p < \infty $. and let $ {\phi _r}(z) = \phi (rz),r < 1$ . If $ \phi $ contains a nontrivial inner factor, it is known that $ \phi /{\phi _r}$ is unbounded in $ {H^p}$-norm. We prove that if $ \phi $ is analytic on the closed disc and has no zeros on the open disc, then $ \phi /{\phi _r} \to 1$ in $ {H^p}$, as $ r \to 1$. The same conclusion follows if $ 1/\phi \in {H^\infty }$. We construct an outer function $ \phi $ which is continuous on the closed disc, analytic for $ z \ne 1$, and such that $ \phi /{\phi _r}$ is unbounded in every $ {H^p}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0283206-6
PII: S 0002-9939(1971)0283206-6
Keywords: Inner function, outer function
Article copyright: © Copyright 1971 American Mathematical Society