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ArticleOriginal scientific text

Title

Ideal norms and trigonometric orthonormal systems

Authors 1

Affiliations

  1. Mathematical Institute, FSU Jena, 07740 Jena, Germany

Abstract

We characterize the UMD-property of a Banach space X by sequences of ideal norms associated with trigonometric orthonormal systems. The asymptotic behavior of those numerical parameters can be used to decide whether X is a UMD-space. Moreover, if this is not the case, we obtain a measure that shows how far X is from being a UMD-space. The main result is that all described sequences are not only simultaneously bounded but are also asymptotically equivalent.

Bibliography

  1. J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 22 (1983), 163-168.
  2. D. L. Burkholder, Martingales and Fourier analysis in Banach spaces, in: Probability and Analysis (Varenna, Italy, 1985), Lecture Notes in Math. 1206, Springer, 1986, 61-108.
  3. M. Defant, Zur vektorwertigen Hilberttransformation, Ph.D. thesis, Christian-Albrechts-Universität Kiel, 1986.
  4. N. Dinculeanu, Vector Measures, Deutscher Verlag der Wiss., Berlin, 1966.
  5. A. Pietsch, Operator Ideals, Deutscher Verlag der Wiss., Berlin, 1978.
  6. A. Pietsch, Eigenvalues and s-numbers, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987.
  7. A. Pietsch and J. Wenzel, Orthogonal systems and geometry of Banach spaces, in preparation.
  8. A. Zygmund, Trigonometric Series, 2nd ed., Cambridge University Press, Cambridge, 1959.
Studia Mathematica
1994-1995/112/1
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Pages:
59-74
Main language of publication
English
Received
1993-12-27
Accepted
1994-04-19
Published
1994
Exact and natural sciences