Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Point evaluations and polynomial approximation in the mean with respect to harmonic measure


Author: John Akeroyd
Journal: Proc. Amer. Math. Soc. 105 (1989), 575-581
MSC: Primary 46E15; Secondary 30E10, 30H05
DOI: https://doi.org/10.1090/S0002-9939-1989-0929403-9
MathSciNet review: 929403
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For $ 1 \leq s < \infty $ and crescents$ ^{1}$ $ G$, with harmonic measure $ \omega $, the author examines the collection of bounded point evaluations, $ \operatorname{bpe}\left( {{P^s}\left( \omega \right)} \right)$, (resp. analytic bounded point evaluations, $ \operatorname{abpe}\left( {{P^s}\left( \omega \right)} \right)$) for polynomials with respect to the $ {L^s}\left( \omega \right)$ norm. If the polynomials are dense in the generalized Hardy space $ {H^s}\left( G \right)$, then $ \operatorname{bpe}\left( {{P^s}\left( \omega \right)} \right) = \operatorname{abpe}\left( {{P^s}\left( \omega \right)} \right) = G$ (Theorem 4). If the polynomials are not dense in $ {H^s}\left( G \right)$, then (with a mild restriction on $ \partial G$) $ \operatorname{bpe} \left( {{P^s}\left( \omega \right)} \right) = \operatorname... ...( {{P^s}\left( \omega \right)} \right) = \operatorname{int} ({\bar G^ \wedge })$ (Theorem 7).


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

  • [1] J. Akeroyd, Polynomial approximation in the mean with respect to harmonic measure on crescents, Trans. Amer. Math. Soc. 303 (1987), 193-199. MR 896016 (88f:30051)
  • [2] J. Brennan, Point evaluations, invariant subspaces and approximation in the mean by polynomials, J. Funct. Anal. 34 (1979), 407-420. MR 556263 (81h:46026)
  • [3] Peter L. Duren, Theory of $ {H^P}$-spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [4] T. W. Gamelin, Uniform algebras, Chelsea, New York, 1984.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E15, 30E10, 30H05

Retrieve articles in all journals with MSC: 46E15, 30E10, 30H05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0929403-9
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society