Hostname: page-component-6bf8c574d5-t27h7 Total loading time: 0 Render date: 2025-02-24T06:27:30.134Z Has data issue: false hasContentIssue false
  • English
  • Français

Orderings in locally compact Abelian groups and the theorem of F. and M. Riesz

Published online by Cambridge University Press:  24 October 2008

Edwin Hewitt  and
Shozo Koshi
Edwin Hewitt
Affiliation:
The University of Washington, Seattle W A 98195, USA
Shozo Koshi
Affiliation:
University of Hokkaido, Sapporo 060, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

Background (1·1). Ordered Abelian groups have been studied for nearly a century. Since the early 1950's, it has been recognized that orderings in locally compact Abelian groups can play an important rôle in harmonic analysis on such groups. In this paper we study orderings, especially in topological Abelian groups with either topological or measure-theoretic properties, obtaining nearly a complete classification of such orderings. We then apply these results to determine the limitations of the celebrated theorem of F. and M. Riesz on such groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 3 , May 1983 , pp. 441 - 457
Copyright
Copyright © Cambridge Philosophical Society 1983

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

REFERENCES

(1)Arens, R.A Banach algebra generalization of conformal mappings on the disc. Trans. Amer. Math. Soc. 81 (1956), 501–513.CrossRefGoogle Scholar
(2)Doss, R.On measures with small Fourier-Stieltjes transforms. Pacific J. Math. 26 (1968), 257–263.Google Scholar
(3)Eedös, J.On the structure of ordered real vector spaces. Publ. Math. Debrecen 4 (1955), 334–343.Google Scholar
(4)Fuchs, L.Abelian groups (Budapest, Publ. House of the Hungarian Academy of Sciences, 1958; Oxford, New York, Pergamon Press, 1960).Google Scholar
(5)Fuchs, L.Partially ordered algebraic systems (Oxford, London, New York, Paris, Pergamon Press, 1963).Google Scholar
(6)Helson, H. and Lowdenslager, D.Prediction theory and Fourier series in several variables. Acta Math. 99 (1958), 165–202.CrossRefGoogle Scholar
(7)Hewitt, E. and Ritter, G.Conjugate Fournier series on certain solenoids. Trans. Amer. Math. Soc. 276 (1983), 817–840.CrossRefGoogle Scholar
(8)Hewitt, E. and Ross, K. A.Abstract harmonic analysis, vol. I, 2nd ed. (Berlin, Heidelberg, New York, Springer-Verlag, 1979).CrossRefGoogle Scholar
(9)Hewitt, E. and Ross, K. A.Abstract harmonic analysis, vol. II (New York, Heidelberg, Berlin, Springer-Verlag, 1970).Google Scholar
(10)Hewitt, E. and Zuckerman, H. S.Singular measures with absolutely continuous convolution squares. Proc. Cambridge Phil. Soc. 62 (1966), 399–420. Corrigendum, Proc. Cambridge Phil. Soc. 63 (1967), 367–368.CrossRefGoogle Scholar
(11)Koshi, S. and Yamaguchi, H.The F. and M. Riesz theorem and group structures. Hokkaido Math. J. 8 (1979), no. 2, 294–299.CrossRefGoogle Scholar
(12)Riesz, F. and Riesz, M. Über die Randwerte einer analytischen Funktion. Comptes Rendus du 4. Congrès dee mathématiciens Scandinaves, Stockholm, 1916, pp. 27–44. Also in Gesammelte Arbeiten Friedrich Riesz, (Budapest, Akadémiai Kiadó, 1960) vol. I, pp. 537–554.Google Scholar
(13)Rudin, W.Fourier analysis on groups (New York, London, John Wiley, 1962).Google Scholar