Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T01:06:47.710Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous singular measures equivalent to their convolution squares

Published online by Cambridge University Press:  24 October 2008

Gavin Brown  and
Edwin Hewitt
Gavin Brown
Affiliation:
University of Liverpool and University of Washington, Seattle
Edwin Hewitt
Affiliation:
University of Liverpool and University of Washington, Seattle
Get access Rights & Permissions [Opens in a new window]

Extract

Throughout this paper, G will denote a locally compact, non-discrete, Abelian group (subjected to various conditions) and X wi11 denote the character group of G. All terminology and notation are as in (7). The measure algebra M(G), as is known, is a very complicated entity. We address ourselves here to some novel peculiarities of the subspace Ms(G) of continuous measures in M(G) that are singular with respect to Haar measure λ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 2 , September 1976 , pp. 249 - 268
Copyright
Copyright © Cambridge Philosophical Society 1976

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)Brown, G.Riesz products and generalized characters. Proc. London Math. Soc. 30 (1975), 209–238.CrossRefGoogle Scholar
(2)Brown, G. & Moran, W.On orthogonality of Riesz products. Proc. Cambridge Philos. Soc. 76 (1974), 173–181.CrossRefGoogle Scholar
(3)Dunford, N. & Schwartz, J. T.Linear operators. Part II: Spectral theory (Interscience Publishers, 1963).Google Scholar
(4)Dunkl, C. F. and Ramirez, D. E.Homomorphisms on groups and induced maps on certain algebras of measures. Trans. Amer. Math. Soc. 160 (1971), 475–485.CrossRefGoogle Scholar
(5)Fomin, S.[ΦOMиH, C.]. On dynamical systems in a space of functions [О динамических системах в пространстве Φункций]. Ukrain. Mat. . 2 (1950), no. 2, 25–47.Google Scholar
(6)Girsanov, I. V.[Гирсанов, И. B.]. On the spectra of dynamical systems generated by stationary Gaussian processes [О спектрах динамических систем, порождаемыхстационарными гауссовскими процессами]. Dokl. Akad. Nauk S.S.S.R. 119 (1958), 851–853.Google Scholar
(7)Hewitt, E. and Ross, K. A.Abstract harmonic analysis, vols. I and II (Springer-Verlag, 1963, 1970).Google Scholar
(8)Hewitt, E. and Stromberg, K. R.Real and abstract analysis. (Springer-Verlag, 1965).Google Scholar
(9)Hewitt, E. & Zuckerman, H. S.Singular measures with absolutely continuous convolution squares. Proc. Cambridge Philos Soc. 62 (1966), 399–420. Corrigendum, Dokl. Akad. Nauk S.S.S.R. 63 (1967), 367–368.Google Scholar
(10)Itô, K.Complex multiple Wiener integral. Japan J. Math. 22 (1952), 63–86.CrossRefGoogle Scholar
(11)Kahane, J.-P. and Salem, R.Ensembles parfaits et séries trigonométriques (Hermann et Cie., Actualités Sci. et Ind. no. 1301, 1963).Google Scholar
(12)Littlewood, J. E.On the Fourier coefficients of functions of bounded variation. Quarterly J. Math. 7 (1936), 219–226.CrossRefGoogle Scholar
(13)Makarov, B. M.[Макаров, Б. М.]. An example of a singular measure equivalent to its convolution square [Пимер сингулярной меры, ∂квивалентной своему квадрату по свертке]. Vestnik Leningrad. Univ. 7 (1968), 51–54.Google Scholar
(14)Salem, R.Algebraic numbers and Fourier analysis (D. C. Heath, 1963).Google Scholar
(15)Sina, J. G.Theory of dynamical systems, Part I (Matematisk Institut, Aarhus, Denmark, Lecture Notes Series, no. 23, 1970).Google Scholar
(16)Stromberg, K. R.Large families of singular measures having absolutely continuous convolution squares. Proc. Cambridge Philos. Soc. 64 (1968), 1015–1022.CrossRefGoogle Scholar