Recovery of $H^{p}$-functions

Author:
V. Totik

Journal:
Proc. Amer. Math. Soc. **90** (1984), 531-537

MSC:
Primary 30D55

DOI:
https://doi.org/10.1090/S0002-9939-1984-0733401-3

MathSciNet review:
733401

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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.

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Keywords:
<IMG WIDTH="33" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${H^p}$"> spaces,
disc algebra,
approximation

Article copyright:
© Copyright 1984
American Mathematical Society