Point evaluations and polynomial approximation in the mean with respect to harmonic measure
HTML articles powered by AMS MathViewer
- by John Akeroyd PDF
- Proc. Amer. Math. Soc. 105 (1989), 575-581 Request permission
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 $\partial G$) $\operatorname {bpe} \left ( {{P^s}\left ( \omega \right )} \right ) = \operatorname {abpe} \left ( {{P^s}\left ( \omega \right )} \right ) = \operatorname {int} ({\bar G^ \wedge })$ (Theorem 7).References
- John Akeroyd, Polynomial approximation in the mean with respect to harmonic measure on crescents, Trans. Amer. Math. Soc. 303 (1987), no. 1, 193–199. MR 896016, DOI 10.1090/S0002-9947-1987-0896016-X
- James E. Brennan, Point evaluations, invariant subspaces and approximation in the mean by polynomials, J. Functional Analysis 34 (1979), no. 3, 407–420. MR 556263, DOI 10.1016/0022-1236(79)90084-3
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655 T. W. Gamelin, Uniform algebras, Chelsea, New York, 1984.
Additional Information
- © Copyright 1989 American Mathematical Society
- 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