Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-06-09T00:58:30.761Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on Riesz sets and lacunary sets

Part of: Abstract harmonic analysis Lie groups

Published online by Cambridge University Press:  09 April 2009

R. G. M. Brummelhuis
R. G. M. Brummelhuis
Affiliation:
Department of Mathematics, University of AmsterdamPlantage Muidergracht 24 1018 TV Amsterdam The Netherlands
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

W. Rudin has proved that the union of the Riesz set N ⊆ R with a Λ(l)-subset of Z is again a Riesz set. In this note we generalize his result to compact groups whose contains a circle group, thereby extending an earlier F. and M. Riesz theorem for such groups by the author. We also investigate the possibility of constructing Λ(p)-sets for these groups, departing from Λ(p)-sets for the circle group in center.

MSC classification

Secondary: 43A05: Measures on groups and semigroups, etc. 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) 22E30: Analysis on real and complex Lie groups 43A77: Analysis on general compact groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 1 , February 1990 , pp. 57 - 65
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Brummelthuis, R. G. M., ‘An F. and M. Riesz theorem for bounded symmetric domains’, Ann. Inst. Fourier 37 (2) (1987), 139–150.CrossRefGoogle Scholar
[2]Cecchini, C., ‘Lacunary Fourier series on compact Lie groups’, J. Funct. Anal. 11 (1972), 191–203.CrossRefGoogle Scholar
[3]Clerc, J. L., ‘Sommes de Riesz et multiplicateurs sur un groupe de Lie compact’, Ann. Inst. Fourier 24 (1) (1974), 149–172.CrossRefGoogle Scholar
[4]Dooley, A. H., ‘Random series for central functions on compact Lie groups’, Illinois J. Math. 24 (4), 545–553.Google Scholar
[5]Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, (Cambridge University Press, 1934).Google Scholar
[6]Hewitt, E. and Ross, K. A., Abstract harmonic analysis, vol. 2 (Springer, New York, 1972).Google Scholar
[7]Koosis, P., Introduction to Hp Spaces, (London Mathematical Society Lecture Note Series 40, Cambridge University Press, 1980).Google Scholar
[8]Price, J. F., ‘Non a sono insiemi infiniti di tipo Λ(p) per SU(2)‘, Boll. Un. Mat. Ital. 4 (4) (1971), 879–881.Google Scholar
[9]Rider, D., ‘Norms of characters and central Λ(p) sets for U(n)’, (Conference on harmonic analysis, College Park, Maryland, SLNM 266, 287–294).Google Scholar
[10]Rider, D., ‘SU(n) has no infinite local Λp sets’, Boll. Un. Mat. Ital. 12 (4) (1975), 155–160.Google Scholar
[11]Rudin, W., ‘Trigonometric series with gaps’, J. Math. Mech. 9 (2) (1960), 203–227.Google Scholar
[12]Wallach, N. R., Harmonic analysis on homogeneous spaces, (Marcel Dekker, New York, 1973).Google Scholar