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On orthogonality of Riesz products

Published online by Cambridge University Press:  24 October 2008

Gavin Brown  and
William Moran
Gavin Brown
Affiliation:
Department of Pure Mathematics, University of Liverpool
William Moran
Affiliation:
Department of Pure Mathematics, University of Liverpool
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Extract

A typical Riesz product on the circle is the weak* limit

where – 1 ≤ rk ≤ 1, øk ∈ R, λT is Haar measure, and the positive integers nk satisfy nk+1/nk ≥ 3. A classical result of Zygmund (11) implies that either µ is absolutely continuous with respect to λT (when ) or µ is purely singular (when ).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 1 , July 1974 , pp. 173 - 181
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Bailey, W. J., Brown, G. and Moran, W.Spectra of independent power measures. Proc. Cambridge Philos. Soc. 72 (1972), 27–35.CrossRefGoogle Scholar
(2)Brown, G. An application of Riesz products. Proceedings of a seminar on Uniform Algebras, March 1973 (Univ. of Aberdeen), pp. 41–47.Google Scholar
(3)Brown, G.Riesz products and generalized characters, Proc. London Math. Soc. To appear.Google Scholar
(4)Brown, G. and Moran, W.A dichotomy for infinite convolutions of discrete measures. Proc. Cambridge Philos. Soc. 73 (1973), 307–16.CrossRefGoogle Scholar
(5)Hewitt, E. and Zuckerman, H. S.Singular measures with absolutely continuous squares. Proc. Cambridge Philos. Soc. 62 (1966), 399–420CrossRefGoogle Scholar
Hewitt, E. and Zuckerman, H. S.Singular measures with absolutely continuous squares. Proc. Cambridge Philos. Soc. 63 (1967), 367–368.CrossRefGoogle Scholar
(6)Jessen, B. and Wintner, A.Distribution functions and the Riemann zeta function. Trans. Amer. Math. Soc. 38 (1935), 48–88.CrossRefGoogle Scholar
(7)Kakutani, S.On equivalence of infinite product measures. Ann. of Math. 49 (1948), 214–224.CrossRefGoogle Scholar
(8)Peyeière, J.Sur les produits de Riesz. C.R. Acad. Sci. Paris Sér. A–B 276 (1973), 1417–1419.Google Scholar
(9)Padé, O.Sur le spectre d'une classe de produits de Riesz. C.R. Acad. Sci. Paris Sér. A–B 276 (1973), 1453–1455.Google Scholar
(10)Taylor, J. L.Ideal theory and Laplace transforms for a class of measure algebras on a group. Acta Math. 121 (1968), 251–292.CrossRefGoogle Scholar
(11)Zygmund, A.On lacunary trigonometric series. Trans. Amer. Math. Soc. 34 (1932), 435–446.CrossRefGoogle Scholar