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Singular measures with absolutely continuous convolution squares

Published online by Cambridge University Press:  24 October 2008

Edwin Hewitt  and
Herbert S. Zuckerman
Edwin Hewitt
Affiliation:
University of Washington, Seattle, U.S.A.
Herbert S. Zuckerman
Affiliation:
University of Washington, Seattle, U.S.A.
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Extract

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such that

for every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 3 , July 1966 , pp. 399 - 420
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

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