Ideal norms and trigonometric orthonormal systems

Jörg Wenzel

Studia Mathematica (1994)

  • Volume: 112, Issue: 1, page 59-74
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Wenzel, Jörg. "Ideal norms and trigonometric orthonormal systems." Studia Mathematica 112.1 (1994): 59-74. <http://eudml.org/doc/216137>.

@article{Wenzel1994,
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.},
author = {Wenzel, Jörg},
journal = {Studia Mathematica},
keywords = {UMD-property; trigonometric orthonormal systems; asymptotically equivalent},
language = {eng},
number = {1},
pages = {59-74},
title = {Ideal norms and trigonometric orthonormal systems},
url = {http://eudml.org/doc/216137},
volume = {112},
year = {1994},
}

TY - JOUR
AU - Wenzel, Jörg
TI - Ideal norms and trigonometric orthonormal systems
JO - Studia Mathematica
PY - 1994
VL - 112
IS - 1
SP - 59
EP - 74
AB - 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.
LA - eng
KW - UMD-property; trigonometric orthonormal systems; asymptotically equivalent
UR - http://eudml.org/doc/216137
ER -

References

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  1. [1] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 22 (1983), 163-168. Zbl0533.46008
  2. [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. [3] M. Defant, Zur vektorwertigen Hilberttransformation, Ph.D. thesis, Christian-Albrechts-Universität Kiel, 1986. Zbl0614.46060
  4. [4] N. Dinculeanu, Vector Measures, Deutscher Verlag der Wiss., Berlin, 1966. 
  5. [5] A. Pietsch, Operator Ideals, Deutscher Verlag der Wiss., Berlin, 1978. 
  6. [6] A. Pietsch, Eigenvalues and s-numbers, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987. Zbl0615.47019
  7. [7] A. Pietsch and J. Wenzel, Orthogonal systems and geometry of Banach spaces, in preparation. Zbl0919.46001
  8. [8] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge University Press, Cambridge, 1959. Zbl0085.05601

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