Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T14:48:22.710Z Has data issue: false hasContentIssue false
  • English
  • Français

Independent sets and lacunarity for hypergroups

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Richard C. Vrem
Richard C. Vrem
Affiliation:
Humboldt State UniversityArcata, California 95521, U.S.A.
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.

Sets of independence are studied for compact abelian hypergroups and they are used, along with Riesz products, to investigate lacunarity questions on the dual object. It is shown that bounded Stechkin sets are always Sidon and that every bounded infinite subset of the dual contains an infinite Sidon set which is also a Λ set. Independent sets are shown to always be Sidon and a necessary condition for Sidonicity is provided. A result of Pisier is used to show that for compact non-abelian groups Sidon and central Λ are equivalent. Several applications are provided, primarily to questions regarding lacunarity on compact groups.

MSC classification

Secondary: 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) 43A40: Character groups and dual objects
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 2 , April 1991 , pp. 171 - 188
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Benke, G., ‘On the hypergroup structure of central Λ(p)-sets’, Pacific J. Math. 50 (1974), 19–27.CrossRefGoogle Scholar
[2]Bloom, W. R., ‘Infinitely divisible measures on hypergroups’, Probability measures on groups, Proc. Conf. Oberwolfach Math. Res. Inst. (Oberwolfach, 1981), pp. 1–15 (Lecture Notes in Mathematics 928, Springer, Berlin, Heidelberg, New York, 1982).Google Scholar
[3]Cartwright, D. I. and McMullen, J. R., ‘A structural criterion for the existence of infinite Sidon sets’, Pacific J. Math. 97 (1981), 301–318.CrossRefGoogle Scholar
[4]Cecchini, C., ‘Lacunary Fourier series on compact Lie groups’, J. Funct. Anal. 11 (1972), 191–203.CrossRefGoogle Scholar
[5]Dunkl, C. F. and Ramirez, D. E., ‘Central Sidon sets of bounded representation type’, Notices Amer. Math. Soc. 18A (1971), 1101.Google Scholar
[6]Dunkl, C. F. and Ramirez, D. E., ‘A family of countable compact P*-hypergroups’, Trans. Amer. Math. Soc. 202 (1975), 339–356.Google Scholar
[7]Hewitt, E. and Ross, K. A., Abstract harmonic analysis II, (Springer-Verlag, New York, 1970).Google Scholar
[8]Hutchinson, M. F., ‘Non-tall compact groups admit infinite Sidon sets’, J. Austral. Math. Soc. 23 (1977), 467–475.CrossRefGoogle Scholar
[9]Jewett, R. I., ‘Spaces with an abstract convolution of measures’, Adv. in Math. 18 (1975),1–101.CrossRefGoogle Scholar
[10]Lasser, R., ‘Orthogonal polynomials and hypergroups’, Rend. Mat. (7) 3 (1983), 185–209.Google Scholar
[11]Lopez, J. M. and Ross, K. A., Sidon sets, (Marcel Dekker, New York, 1975).Google Scholar
[12]Marcus, M. B. and Pisier, G., Random Fourier series with applications to harmonic analysis, (Ann. of Math Studies, 101, Princeton Univ. Press, 1981).Google Scholar
[13]McMullen, J.R., ‘On the dual object of a compact connected group’, Math. Z. 185 (1984), 539–552.CrossRefGoogle Scholar
[14]Parker, W. A., ‘Central Sidon sets and central Λ(p)-sets’, J. Austral. Math.Soc. 24 (1972), 62–74.CrossRefGoogle Scholar
[15]Pisier, G., ‘Lacunarité et processus gaussiens’, C.R. Acad. Sci. Paris Ser. A 286 (1978), 1003–1006.Google Scholar
[16]Pisier, G., ‘De nouvelles caractérisations des ensembles de Sidon’, Mathematical Analysis and Applications, Part B, pp. 685–726 (Advances in Mathematics Supplementary Studies, 7B, 1981).Google Scholar
[17]Pisier, G., ‘Arithmetic characterizations of Sidon sets’, Bull. Amer. Math. Soc. 8 (1983), 87–89.CrossRefGoogle Scholar
[18]Rider, D., ‘Gap series on groups and spheres’, Canad. J.Math. 18 (1966),389–398.CrossRefGoogle Scholar
[19]Rider, D., ‘Central lacunary sets’, Monatsh.Math. 76 (1972), 328–338.CrossRefGoogle Scholar
[20]Ross, K. A., ‘Centers of hypergroups’, Trans. Amer. Math. Soc. 243 (1978), 251–269.CrossRefGoogle Scholar
[21]Rudin, W., Fourier analysis on groups, (Interscience Tracts in Pure and Appl.Math., no. 12, Interscience, New York, 1962).Google Scholar
[22]Vrem, R. C., ‘Lacunarity on compact hypergroups’, Math. Z. 164 (1978), 93–104.CrossRefGoogle Scholar
[23]Vrem, R. C., ‘Continuous measures and lacunarity on hypergroups’, Trans. Amer. Math. Soc. 269 (1982), 549–556.CrossRefGoogle Scholar
[24]Wilson, D. C., ‘Lacunarity for tall compact groups: Figà-Talamanca-Rider sets’, Monatsh. Math. 101 (1986), 67–74.CrossRefGoogle Scholar