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

N. Kalton

Colloquium Mathematicae (1993)

  • Volume: 64, Issue: 2, page 233-244
  • ISSN: 0010-1354

Abstract

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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 L p -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 ( I 0 -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.

How to cite

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Kalton, N.. "On vector-valued inequalities for Sidon sets and sets of interpolation." Colloquium Mathematicae 64.2 (1993): 233-244. <http://eudml.org/doc/210187>.

@article{Kalton1993,
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 $L_p$-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 ($I_0$-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.},
author = {Kalton, N.},
journal = {Colloquium Mathematicae},
keywords = {compact Abelian group; quasi-Banach space; Rademacher function; Borel functions; Sidon set; Banach space; non-locally convex spaces; set of interpolation; quasinorm},
language = {eng},
number = {2},
pages = {233-244},
title = {On vector-valued inequalities for Sidon sets and sets of interpolation},
url = {http://eudml.org/doc/210187},
volume = {64},
year = {1993},
}

TY - JOUR
AU - Kalton, N.
TI - On vector-valued inequalities for Sidon sets and sets of interpolation
JO - Colloquium Mathematicae
PY - 1993
VL - 64
IS - 2
SP - 233
EP - 244
AB - 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 $L_p$-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 ($I_0$-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.
LA - eng
KW - compact Abelian group; quasi-Banach space; Rademacher function; Borel functions; Sidon set; Banach space; non-locally convex spaces; set of interpolation; quasinorm
UR - http://eudml.org/doc/210187
ER -

References

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  1. [1] N. Asmar and S. J. Montgomery-Smith, On the distribution of Sidon series, Ark. Mat., to appear. Zbl0836.43011
  2. [2] D. Grow, A class of I 0 -sets, Colloq. Math. 53 (1987), 111-124. 
  3. [3] S. Hartman and C. Ryll-Nardzewski, Almost periodic extensions of functions, ibid. 12 (1964), 23-39. Zbl0145.32101
  4. [4] S. Hartman and C. Ryll-Nardzewski, Almost periodic extensions of functions, II, ibid. 15 (1966), 79-86. 
  5. [5] J.-P. Kahane, Ensembles de Ryll-Nardzewski et ensembles de Helson, ibid. 15 (1966), 87-92. Zbl0144.34202
  6. [6] N. J. Kalton, Banach envelopes of non-locally convex spaces, Canad. J. Math. 38 (1986), 65-86. Zbl0577.46016
  7. [7] N. J. Kalton, Plurisubharmonic functions on quasi-Banach spaces, Studia Math. 84 (1986), 297-324. Zbl0625.46021
  8. [8] J.-F. Méla, Sur les ensembles d'interpolation de C. Ryll-Nardzewski et de S. Hartman, ibid. 29 (1968), 167-193. Zbl0155.18802
  9. [9] J.-F. Méla, Sur certains ensembles exceptionnels en analyse de Fourier, Ann. Inst. Fourier (Grenoble) 18 (2) (1968), 32-71. Zbl0187.07202
  10. [10] J. Mycielski, On a problem of interpolation by periodic functions, Colloq. Math. 8 (1961), 95-97. Zbl0102.05302
  11. [11] A. Pełczyński, Commensurate sequences of characters, Proc. Amer. Math. Soc. 104 (1988), 525-531. Zbl0693.46044
  12. [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. [13] C. Ryll-Nardzewski, Concerning almost periodic extensions of functions, Colloq. Math. 12 (1964), 235-237. Zbl0133.04303
  14. [14] E. Strzelecki, Some theorems on interpolation by periodic functions, ibid. 12 (1964), 239-248. Zbl0133.02003

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