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Commensurate sequences of characters

Author: A. Pełczyński
Journal: Proc. Amer. Math. Soc. 104 (1988), 525-531
MSC: Primary 43A15; Secondary 43A46, 46B15
MathSciNet review: 962823
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Abstract: If $ ({a_j})$ and $ ({b_j})$ are sequences of characters on compact abelian groups $ S$ and $ T$ respectively such that for every sequence of scalars $ ({\alpha _j})\vert\vert\sum {\alpha _j}{a_j}\vert{\vert _\infty } \asymp \vert\vert\sum {\alpha _j}{b_j}\vert{\vert _\infty }$ tnen for every $ 1 \leq p < \infty $ and every sequence $ ({x_j})$ of elements of an arbitrary Banach space $ X$

$\displaystyle {\int_S {\left\Vert {\sum {{x_j}{a_j}} } \right\Vert} ^p}ds \asymp {\int_T {\left\Vert {\sum {{x_j}b} } \right\Vert} ^p}dt.$

This result generalizes a result of Pisier [Pi 1] for Sidon sets. For topological Sidon sets on $ {\mathbf{R}}$ a slightly stronger result holds.

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

  • [DG 1] Myriam Déchamps-Gondim, Sur les ensembles de Sidon topologiques, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A1247–A1249 (French). MR 0275065
  • [Dg 2] Myriam Déchamps-Gondim, Ensembles de Sidon topologiques, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 3, 51–79 (French, with English summary). MR 0340981
  • [Kh] Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
  • [M] Yves Meyer, Algebraic numbers and harmonic analysis, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1972. North-Holland Mathematical Library, Vol. 2. MR 0485769
  • [O] W. Orlicz, Über unbedingte Konvergenz in Funktionräumen. I, II, Studia Math 4 (1933), 33-37, 41-47.
  • [Pi 1] G. Pisier, Les inégalités de Khintchine-Kahane, d’après C. Borell, Séminaire sur la Géométrie des Espaces de Banach (1977–1978), École Polytech., Palaiseau, 1978, pp. Exp. No. 7, 14 (French). MR 520209
  • [Pi 2] Gilles Pisier, Ensembles de Sidon et processus gaussiens, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 15, A671–A674 (French, with English summary). MR 0511046
  • [Pi 3] -, De nouvelles caractérisation des ensembles de Sidon, Math Anal. Appl., Part B, Advances in Math. Suppl. Studies 7B (1981).
  • [R] Walter Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203–227. MR 0116177
  • [Z] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776

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Keywords: Commensurate sequences, Sidon sets
Article copyright: © Copyright 1988 American Mathematical Society