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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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.
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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