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Function theory in spaces
of uniformly convergent Fourier series

Author: Scott F. Saccone
Journal: Proc. Amer. Math. Soc. 128 (2000), 1813-1823
MSC (1991): Primary 46E15, 32C15
Published electronically: October 29, 1999
MathSciNet review: 1664367
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Abstract: We study spaces of continuous functions on the unit circle with uniformly convergent Fourier series and show they possess such Banach space properties as the Pelczynski property, the Dunford-Pettis property and the weak sequential completeness of the dual space. We also prove extensions of theorems of Mooney and Sarason from the Hardy space $H^{\scriptscriptstyle \infty }$ to the space $H^{\scriptscriptstyle \infty }_{\scriptscriptstyle U}$ of bounded analytic functions whose partial Fourier sums are uniformly bounded.

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

  • [1] J. Bourgain, Quelques proprietes lineaires topologiques de l'espace des series de Fourier uniformement convergentes, Seminaire Initiation a l'Analyse, G. Choquet, M. Rogalski, J. Saint Raymond, 22e annee. Univ. Paris-6, 1982-83, Expose no. 14.
  • [2] -, On weak completeness of the dual of spaces of analytic and smooth functions, Bull. Soc. Math. Belg. Serie B 35 (1) (1983), 111-118. MR 84j:46039
  • [3] -, The Dunford-Pettis property for the ball-algebras, the polydisc algebras and the Sobolev spaces, Studia Math. 77 (3) (1984), 245-253. MR 85f:46044
  • [4] B.J. Cole and T.W. Gamelin, Tight uniform algebras and algebras of analytic functions, J. Funct. Anal. (46) (1982), 158-220. MR 83h:46065
  • [5] T.W. Gamelin, Uniform Algebras, Chelsea Publishing Co., New York, 1984. MR 53:14137 (1969 edition)
  • [6] S.V. Hru\v{s}\v{c}ev and S.A. Vinogradov, Free interpolation in the space of uniformly convergent Taylor series, Lecture Notes in Math, 864, Complex Analysis and Spectral Theory, Ed. V.P. Havin and N.K. Nikol'skii, 1981, 143-171. MR 82m:46001
  • [7] J. Kahane and K. Katznelson, Sur les séries de Fourier uniformément convergentes, C.R. Acad. Sc. Paries, t.261 (18 octobre 1965) Groupe 1, 3025-3028. MR 32:6131
  • [8] M.C. Mooney, A theorem on bounded analytic functions, Pac. J. Math. 43 (1973), 457-463. MR 47:2374
  • [9] D. Oberlin, A Rudin-Carleson theorem for uniformly convergent Fourier series, Mich. Math. J. 27 (1980), 309-313. MR 81j:46082
  • [10] S.F. Saccone, Banach space properties of strongly tight uniform algebras, Studia Math. 114 (2) (1995), 159-180. MR 96d:46068
  • [11] -, The Pe{\l}czy\'{n}ski property for tight subspaces, J. Funct. Anal. 148 (1) (1997), 86-116. MR 98i:46014
  • [12] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, New York, 1991. MR 93d:46001

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Additional Information

Scott F. Saccone
Affiliation: Department of Mathematics, Campus Box 1146, Washington University, Saint Louis, Missouri 63130
Address at time of publication: Department of Mathematics, 202 Mathematical Sciences Building, The University of Missouri, Columbia, Missouri 65211

Received by editor(s): August 4, 1998
Published electronically: October 29, 1999
Additional Notes: The author was partially supported by National Science Foundation grant DMS 9705851.
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

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