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

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).