Download PDF - On vector-valued inequalities for Sidon sets and sets of interpolationAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

On vector-valued inequalities for Sidon sets and sets of interpolation

Authors 1

Affiliations

  1. Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211, U.S.A.

Abstract

Let E be a Sidon subset of the integers and suppose X is a Banach space. Then Pisier has shown that E-spectral polynomials with values in X behave like Rademacher sums with respect to Lp-norms. We consider the situation when X is a quasi-Banach space. For general quasi-Banach spaces we show that a similar result holds if and only if E is a set of interpolation (I0-set). However, for certain special classes of quasi-Banach spaces we are able to prove such a result for larger sets. Thus if X is restricted to be "natural" then the result holds for all Sidon sets. We also consider spaces with plurisubharmonic norms and introduce the class of analytic Sidon sets.

Bibliography

  1. N. Asmar and S. J. Montgomery-Smith, On the distribution of Sidon series, Ark. Mat., to appear.
  2. D. Grow, A class of I0-sets, Colloq. Math. 53 (1987), 111-124.
  3. S. Hartman and C. Ryll-Nardzewski, Almost periodic extensions of functions, ibid. 12 (1964), 23-39.
  4. S. Hartman and C. Ryll-Nardzewski, Almost periodic extensions of functions, II, ibid. 15 (1966), 79-86.
  5. J.-P. Kahane, Ensembles de Ryll-Nardzewski et ensembles de Helson, ibid. 15 (1966), 87-92.
  6. N. J. Kalton, Banach envelopes of non-locally convex spaces, Canad. J. Math. 38 (1986), 65-86.
  7. N. J. Kalton, Plurisubharmonic functions on quasi-Banach spaces, Studia Math. 84 (1986), 297-324.
  8. J.-F. Méla, Sur les ensembles d'interpolation de C. Ryll-Nardzewski et de S. Hartman, ibid. 29 (1968), 167-193.
  9. J.-F. Méla, Sur certains ensembles exceptionnels en analyse de Fourier, Ann. Inst. Fourier (Grenoble) 18 (2) (1968), 32-71.
  10. J. Mycielski, On a problem of interpolation by periodic functions, Colloq. Math. 8 (1961), 95-97.
  11. A. Pełczyński, Commensurate sequences of characters, Proc. Amer. Math. Soc. 104 (1988), 525-531.
  12. G. Pisier, Les inégalités de Kahane-Khintchin d'après C. Borell, in: Séminaire sur la géométrie des espaces de Banach, Ecole Polytechnique, Palaiseau, Exposé VII, 1977-78.
  13. C. Ryll-Nardzewski, Concerning almost periodic extensions of functions, Colloq. Math. 12 (1964), 235-237.
  14. E. Strzelecki, Some theorems on interpolation by periodic functions, ibid. 12 (1964), 239-248.
Colloquium Mathematicum
1993/64/2
Export to PBN, Opens in new tab
Pages:
233-244
Main language of publication
English
Received
1992-03-17
Published
1993
Exact and natural sciences