Continuity of extremal elements in uniformly convex spaces

Ferguson, Timothy

doi:10.1090/S0002-9939-09-09892-X

Continuity of extremal elements in uniformly convex spaces
HTML articles powered by AMS MathViewer

by Timothy Ferguson

Proc. Amer. Math. Soc. 137 (2009), 2645-2653

DOI: https://doi.org/10.1090/S0002-9939-09-09892-X

Published electronically: March 17, 2009

PDF

Abstract:

In this paper, we study the problem of finding the extremal element for a linear functional over a uniformly convex Banach space. We show that a unique extremal element exists and depends continuously on the linear functional, and vice versa. Using this, we simplify and clarify Ryabykh’s proof that for any linear functional on a uniformly convex Bergman space with kernel in a certain Hardy space, the extremal function belongs to the corresponding Hardy space.

References

Dov Aharonov, Catherine Bénéteau, Dmitry Khavinson, and Harold Shapiro, Extremal problems for nonvanishing functions in Bergman spaces, Selected topics in complex analysis, Oper. Theory Adv. Appl., vol. 158, Birkhäuser, Basel, 2005, pp. 59–86. MR 2147588, DOI 10.1007/3-7643-7340-7_{5}
James A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396–414. MR 1501880, DOI 10.1090/S0002-9947-1936-1501880-4
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Peter Duren and Alexander Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. MR 2033762, DOI 10.1090/surv/100
Dmitry Khavinson, John E. McCarthy, and Harold S. Shapiro, Best approximation in the mean by analytic and harmonic functions, Indiana Univ. Math. J. 49 (2000), no. 4, 1481–1513. MR 1836538, DOI 10.1512/iumj.2000.49.1787
Dmitry Khavinson and Michael Stessin, Certain linear extremal problems in Bergman spaces of analytic functions, Indiana Univ. Math. J. 46 (1997), no. 3, 933–974. MR 1488342, DOI 10.1512/iumj.1997.46.1375
Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
V. G. Ryabykh, Extremal problems for summable analytic functions, Sibirsk. Mat. Zh. 27 (1986), no. 3, 212–217, 226 (Russian). MR 853902
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
Dragan Vukotić, Linear extremal problems for Bergman spaces, Exposition. Math. 14 (1996), no. 4, 313–352. MR 1418027