Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-19T06:45:23.000Z Has data issue: false hasContentIssue false
  • English
  • Français

Properties of Littlewood-Paley sets

Published online by Cambridge University Press:  04 October 2011

Kathryn E. Hare  and
Ivo Klemes
Kathryn E. Hare
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Ivo Klemes
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 2K6, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a combinatorial characterization of the known examples of intervals of ℤ having the Littlewood-Paley (LP) property. This leaves an interesting open problem. We also note that certain previously known necessary conditions are equivalent, and give two examples of intervals which are not LP but whose endpoints form thin sets (e.g. Sidon or Λ(p) for all p < ∞).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 485 - 494
Copyright
Copyright © Cambridge Philosophical Society 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Edwards, R. E. and Gaudry, G. I.. Littlewood-Paley and Multiplier Theory (Springer-Verlag, 1977).CrossRefGoogle Scholar
[2] Fournier, J. and Pigno, L.. Analytic and arithmetic properties of thin sets. Pacific J. Math. 105 (1983), 115–141.CrossRefGoogle Scholar
[3] Gaudry, G. I.. Littlewood-Paley theorems for sum and difference sets. Math. Proc. Cambridge Philos. Soc. 83 (1978), 65–71.CrossRefGoogle Scholar
[4] Gundy, R. F. and Varopoulous, N. Th.. A martingale that occurs in harmonic analysis. Ark. Mat. 14 (1976), 179–187.CrossRefGoogle Scholar
[5] Lopez, J. and Ross, K.. Sidon Sets. Lecture Notes in Pure and Applied Math. no. 13 (Marcel Dekker Inc., 1975).Google Scholar
[6] Miheev, I. M.. On lacunary series. Math. USSR-Sb. 27 (1975), 481–502.CrossRefGoogle Scholar
[7] Francia, J. L. Rubio De. A Littlewood-Paley inequality for arbitrary intervals. Rev. Mat. Iberoamericana 1 (1985), 1–14.CrossRefGoogle Scholar
[8] Rudin, W.. Trigonometric series with gaps. J. Math. Mech. 9 (1960), 203–227.Google Scholar
[9] Sjögren, P. and Sjölin, P.. Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets. Ann. Inst. Fourier (Grenoble) 31 (1981), 157–175.CrossRefGoogle Scholar