Advertisement

Israel Journal of Mathematics

, Volume 12, Issue 4, pp 356–368 | Cite as

p-Helson sets, 1<p<2

  • Michael B. Gregory
Article
  • 58 Downloads

Abstract

Ap-Helson set is defined to be a closed subsetE of the circle groupT with the property that every continuous function onE can be extended to the full circle in such a way that this extension has its sequence of Fourier coefficients inl p. For 1<p<2, the union of two such sets is again ap-Helson set. It is shown that thep-Helson sets (p>1) differ from the Helson sets and also that the notion really depends on the indexp. An analogue of H. Helson’s result is given: ap-Helson set supports no nonzero measure with Fourier-Stieltjes transform inl q, 1/p+1/q=1.

Keywords

Fourier Coefficient Lebesgue Measure Zero Circle Group Total Variation Norm Pisot Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. K. Bary,A Treatise on Trigonometric Series, Vol. 1, Macmillan, New York, 1964.MATHGoogle Scholar
  2. 2.
    C. C. Graham,Compact independent sets and Haar measure, Proc. Amer. Math. Soc. (to appear).Google Scholar
  3. 3.
    H. Helson,Fourier transforms on perfect sets, Studia Math.14 (1954), 209–213.MathSciNetGoogle Scholar
  4. 4.
    E. Hewitt and K. Ross,Abstract Harmonic Analysis II, Springer-Verlag, New York, 1970.MATHGoogle Scholar
  5. 5.
    J. P. Kahane R. Salem,Ensembles Parfaits et Séries Trigonométriques, Hermann, Paris, 1963.MATHGoogle Scholar
  6. 6.
    J. P. Kahane,Séries de Fourier absolument convergent, Ergebnisse der Mathematik, Band 50, Springer-Verlag, New York, 1970.Google Scholar
  7. 7.
    Y. Katznelson,An Introduction to Harmonic Analysis, Wiley, New York, 1968.MATHGoogle Scholar
  8. 8.
    W. Rudin,Trigonometric series with gaps, J. Math. Mech.9 (1960), 203–228.MATHMathSciNetGoogle Scholar
  9. 9.
    W. Rudin,Fourier Analysis on Groups, Interscience, New York, 1962.MATHGoogle Scholar
  10. 10.
    W. Rudin,Real and Complex Analysis, McGraw-Hill, New York, 1966.MATHGoogle Scholar
  11. 11.
    R. Salem,On sets of multiplicity for trigonometrical series, Amer. J. Math.64 (1942), 531–538.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    N. Th. Varopoulos,Sur la réunion de deux ensembles de Helson,C. R. Acad. Sci. Paris. 271, 251–253.Google Scholar
  13. 13.
    I. Wik,On linear dependence in closed sets, Ark. Mat.4 (1960), 209–218.CrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1972

Authors and Affiliations

  • Michael B. Gregory
    • 1
  1. 1.The University of North DakotaNorth DakotaUSA

Personalised recommendations