Recovery of $H^{p}$-functions

Abstract:

Let there be given finitely many points $\{ {\alpha _k}\} _1^n$ from the unit disc. If $f$ is a ${H^p}$-function then how well can the value of $f$ at $z = 0$ be approximated by linear means $\sum \nolimits _1^n {{c_k}f({\alpha _k})?}$ We give the optimal constants ${c_k}$ and get, as a corollary, the possibility of the approximation of $f$ by operators of the form $\sum \nolimits _1^n {f({\alpha _k}){p_k}}$ with polynomials ${p_k}$. The order of approximation depends on the distance $\sum \nolimits _1^n {\left ( {1 - \left | {{\alpha _k}} \right |} \right )}$ of the point system from the unit circle.