Illinois Journal of Mathematics

Extremal problems in Bergman spaces and an extension of Ryabykh’s theorem

Timothy Ferguson

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We study linear extremal problems in the Bergman space $A^p$ of the unit disc for $p$ an even integer. Given a functional on the dual space of $A^p$ with representing kernel $k \in A^q$, where $1/p + 1/q = 1$, we show that if the Taylor coefficients of $k$ are sufficiently small, then the extremal function $F \in H^{\infty}$. We also show that if $q \le q_1 < \infty$, then $F \in H^{(p-1)q_1}$ if and only if $k \in H^{q_1}$.

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Illinois J. Math., Volume 55, Number 2 (2011), 555-573.

First available in Project Euclid: 1 February 2013

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Zentralblatt MATH identifier

Primary: 30D60: Quasi-analytic and other classes of functions 30D65


Ferguson, Timothy. Extremal problems in Bergman spaces and an extension of Ryabykh’s theorem. Illinois J. Math. 55 (2011), no. 2, 555--573. doi:10.1215/ijm/1359762402.

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  • J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414.
  • P. Duren, Theory of $H\sp{p}$ spaces, Dover, New York, 1970.
  • P. Duren and A. Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004.
  • T. Ferguson, Continuity of extremal elements in uniformly convex spaces, Proc. Amer. Math. Soc. 137 (2009), 2645–2653.
  • J. B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1981.
  • E. Hewitt and K. Stromberg, Real and abstract analysis, 3rd printing, Graduate Texts in Mathematics, vol. 25, Springer-Verlag, New York, 1975.
  • D. Khavinson and M. Stessin, Certain linear extremal problems in Bergman spaces of analytic functions, Indiana Univ. Math. J. 46 (1997), 933–974.
  • V. G. Ryabykh, Extremal problems for summable analytic functions, Sibirsk. Mat. Zh. 27 (1986), 212–217, 226.
  • H. S. Shapiro, Topics in approximation theory, Lecture Notes in Math., vol. 187, Springer-Verlag, Berlin, 1971.
  • D. Vukotić, Linear extremal problems for Bergman spaces, Expo. Math. 14 (1996), 313–352.
  • A. Zygmund, Trigonometric series: Vols. I, II, 2nd ed., Cambridge University Press, London, 1968.