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Proceedings of the American Mathematical Society
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
MathSciNet review: 929403
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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).


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0929403-9
PII: S 0002-9939(1989)0929403-9
Article copyright: © Copyright 1989 American Mathematical Society