Recovery of $H^{p}$-functions
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- by V. Totik PDF
- Proc. Amer. Math. Soc. 90 (1984), 531-537 Request permission
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.References
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- Harold S. Shapiro, Topics in approximation theory, Lecture Notes in Mathematics, Vol. 187, Springer-Verlag, Berlin-New York, 1971. With appendices by Jan Boman and Torbjörn Hedberg. MR 0437981, DOI 10.1007/BFb0058976 G. Somorjai, On discrete linear operators in the function space $A$, Proc. Constructive Function Theory (Blagoevgrad, 1977), Publ. H. Bulgarian Acad. Sci., Sofia, 1980, pp. 489-500. A. F. Timan, Theory of approximation of functions of a real variable, Hindustan, Delhi, 1966.
Additional Information
- © Copyright 1984 American Mathematical Society
- 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