Rational Functions in Bergman Spaces

  • Evsey Dyn’kin
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 113)

Abstract

An inversion of the classical Hardy-Littlewood theorem on embedding of Hardy spaces into Bergman spaces is proved for the subset of rational functions of a given degree. Many other embeddings for such functions follow from this result. For the Hardy-Littlewood theorem itself, a refinement in terms of rational approximation is proved. As an application, a sharp rational approximation theorem in the Bergman norm is obtained.

Keywords

Rational Function Hardy Space Rational Approximation Besov Space Bergman Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Yu. A. Brudnyi, Adaptive approximation of functions with singularities, Trans. Moscow Math. Soc., 55 (1994), 123–186.MathSciNetGoogle Scholar
  2. [2]
    A. P. Calderón, On an integral of Marcinkiewicz, Studia Math., 57 (1976), 279–284.MathSciNetMATHGoogle Scholar
  3. [3]
    P. L. Duren, Theory of H P spaces, Academic Press, New York, 1970.MATHGoogle Scholar
  4. [4]
    E.M. Dyn’kin, On the classes Bp sfor 0 < p < 1, Soviet Math Dokl., 29 No. 2 (1984), 357–360.MATHGoogle Scholar
  5. [5]
    E. M. Dyn’kin, Inequalities for rational functions, J. Approx. Theory, 91 (1997), 349–367.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981.MATHGoogle Scholar
  7. [7]
    J. Garsia-Cuerva and J.-L. Rubio de Francia, Weighted norm inequalities and related topics, North Holland, 1985.Google Scholar
  8. [8]
    A. A. Pekarskii, An inequality of Bernstein type for derivatives of rational functions and inverse theorems of rational approximation, Math. USSR-Sb. 52 (1985), 557–574.CrossRefGoogle Scholar
  9. [9]
    A. A. PekarskiiClasses of analytic functions defined by best rational approximations in HP, Math. USSR-Sb., 55 (1986),1–18.CrossRefGoogle Scholar
  10. [10]
    E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton N. J., 1970.MATHGoogle Scholar

Copyright information

© Springer Basel AG 2000

Authors and Affiliations

  • Evsey Dyn’kin

There are no affiliations available

Personalised recommendations