Abstract
We discuss approximation of extremal functions by polynomials in the weighted Bergman spaces \(A^p_\alpha \) where \(-1< \alpha < \min (0,p-2)\). We obtain bounds on how close the approximation is to the true extremal function in the \(A^p_\alpha \) and uniform norms. We also prove several results on the relation between the Bergman modulus of continuity of a function and how quickly its best polynomial approximants converge to it.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ball, Keith, Carlen, Eric A., Lieb, Elliott H.: Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115(3), 463–482 (1994)
Bénéteau, C., Khavinson, D.: A survey of linear extremal problems in analytic function spaces. In: Complex analysis and potential theory, CRM Proc. Lecture Notes, vol. 55, Amer. Math. Soc., Providence, RI, pp. 33–46 (2012)
Bénéteau, C., Khavinson, D.: Selected problems in classical function theory. In: Invariant subspaces of the shift operator, Contemp. Math., vol. 638, Amer. Math. Soc., Providence, RI, pp. 255–265 (2015)
Clarkson, J.A.: Uniformly convex spaces. Trans. Amer. Math. Soc. 40(3), 396–414 (1936)
Duren, Peter: Theory of \(H^{p}\) spaces, Pure and Applied Mathematics, vol. 38. Academic Press, New York (1970)
Duren, P., Schuster, A.: Bergman spaces, mathematical Surveys and Monographs, vol. 100. American Mathematical Society, Providence (2004)
Ferguson, T.: Continuity of extremal elements in uniformly convex spaces. Proc. Amer. Math. Soc. 137(8), 2645–2653 (2009)
Ferguson, T.: Bergman-Hölder functions, area integral means and extremal problems. Integral Equ Oper Theory 87(4), 545–563 (2017)
Hedenmalm, Håkan, Korenblum, Boris, Zhu, Kehe: Theory of Bergman Spaces, Graduate Texts in Mathematics, vol. 199. Springer, New York (2000)
Khavinson, D., McCarthy, J.E., Shapiro, H.S.: Best approximation in the mean by analytic and harmonic functions. Indiana Univ. Math. J. 49(4), 1481–1513 (2000)
Khavinson, Dmitry, Stessin, Michael: Certain linear extremal problems in Bergman spaces of analytic functions. Indiana Univ. Math. J. 46(3), 933–974 (1997)
Ryabykh, V.G.: Extremal problems for summable analytic functions. Sibirsk. Mat. Zh. 27(3), 212–217 (1986). 226 ((in Russian))
Shapiro, H.S.: Topics in approximation theory. With appendices by Jan Boman and Torbjörn Hedberg, Lecture Notes in Math., vol. 187. Springer, Berlin (1971)
Harold, S.: Shapiro, regularity properties of the element of closest approximation. Trans. Amer. Math. Soc. 181, 127–142 (1973)
Zygmund, A.: Trigonometric series: vols. I, II, Second edn, reprinted with corrections and some additions. Cambridge University Press, London (1968)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dmitry Khavinson.
Partially supported by RGC Grant RGC-2015-22 from the University of Alabama.
Thanks to Brendan Ames for a helpful discussion, and to the referee for helpful comments.
Rights and permissions
About this article
Cite this article
Ferguson, T. Uniform Approximation of Extremal Functions in Weighted Bergman Spaces. Comput. Methods Funct. Theory 18, 439–453 (2018). https://doi.org/10.1007/s40315-017-0230-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40315-017-0230-2