Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

   
 
 

 

Continuity of extremal elements in uniformly convex spaces


Author: Timothy Ferguson
Journal: Proc. Amer. Math. Soc. 137 (2009), 2645-2653
MSC (2000): Primary 30H05; Secondary 46B99
DOI: https://doi.org/10.1090/S0002-9939-09-09892-X
Published electronically: March 17, 2009
MathSciNet review: 2497477
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. 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 (2006i:30047)
  • 2. James A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396-414. MR 1501880
  • 3. Peter Duren, Theory of $ H\sp{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • 4. Peter Duren and Alexander Schuster, Bergman spaces, Mathematical Surveys and Monographs, Vol. 100, American Mathematical Society, Providence, RI, 2004. MR 2033762 (2005c:30053)
  • 5. 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 (2002b:41023)
  • 6. 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 (99k:30080)
  • 7. Gottfried Köthe, Topological vector spaces. I, Translated from the German by D. J. H. Garling, Die Grundlehren der Mathematischen Wissenschaften, Band 159, Springer-Verlag New York Inc., New York, 1969. MR 0248498 (40:1750)
  • 8. V. G. Ryabykh, Extremal problems for summable analytic functions, Sibirsk. Mat. Zh. 27 (1986), no. 3, 212-217, 226 (in Russian). MR 853902 (87j:30058)
  • 9. Harold S. Shapiro, Topics in approximation theory, with appendices by Jan Boman and Torbjörn Hedberg, Lecture Notes in Math., Vol. 187, Springer-Verlag, Berlin, 1971.MR 0437981 (55:10902)
  • 10. Dragan Vukotić, Linear extremal problems for Bergman spaces, Exposition. Math. 14 (1996), no. 4, 313-352. MR 1418027 (97m:46117)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30H05, 46B99

Retrieve articles in all journals with MSC (2000): 30H05, 46B99


Additional Information

Timothy Ferguson
Affiliation: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043
Email: tjferg@umich.edu

DOI: https://doi.org/10.1090/S0002-9939-09-09892-X
Received by editor(s): September 9, 2008
Published electronically: March 17, 2009
Communicated by: Mario Bonk
Article copyright: © Copyright 2009 Timothy Ferguson

American Mathematical Society