A remark on the F. and M. Riesz theorem

Pigno, Louis; Smith, Brent

doi:10.1090/S0002-9939-1980-0572307-6

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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A remark on the F. and M. Riesz theorem
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by Louis Pigno and Brent Smith

Proc. Amer. Math. Soc. 79 (1980), 585-586

DOI: https://doi.org/10.1090/S0002-9939-1980-0572307-6

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Abstract:

Let $\mu$ be a measure of analytic type on the unit circle. We give a short direct proof that ${\mu ^2}$ is absolutely continuous with respect to Lebesgue measure; our method also gives a convolution product version of some related several variable results.

References

Henry Helson, On a theorem of F. and M. Riesz, Colloq. Math. 3 (1955), 113–117. MR 69306, DOI 10.4064/cm-3-2-113-117
Bernt K. Øksendal, A short proof of the F. and M. Riesz theorem, Proc. Amer. Math. Soc. 30 (1971), 204. MR 279316, DOI 10.1090/S0002-9939-1971-0279316-X