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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Conjugate Fourier series on certain solenoids

Authors: Edwin Hewitt and Gunter Ritter
Journal: Trans. Amer. Math. Soc. 276 (1983), 817-840
MSC: Primary 43A70; Secondary 42A50
MathSciNet review: 688979
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Abstract: We consider an arbitrary noncyclic subgroup of the additive group $ {\mathbf{Q}}$ of rational numbers, denoted by $ {{\mathbf{Q}}_{\mathbf{a}}}$, and its compact character group $ {\Sigma _{\mathbf{a}}}$. For $ 1 < p < \infty $, an abstract form of Marcel Riesz's theorem on conjugate series is known. For $ f$ in $ {\mathfrak{L}_p}({\Sigma _{\mathbf{a}}})$, there is a function $ \tilde{f}$ in $ {\mathfrak{L}_p}({\Sigma _{\mathbf{a}}})$ whose Fourier transform $ (\tilde{f})\hat{\empty}(\alpha )$ at $ \alpha $ in $ {{\mathbf{Q}}_{\mathbf{a}}}$ is $ - i\,\operatorname{sgn}\,\alpha \hat{f}(\alpha )$. We show in this paper how to construct $ \tilde{f}$ explicitly as a pointwise limit almost everywhere on $ {\Sigma_{\mathbf{a}}}$ of certain harmonic functions, as was done by Riesz for the circle group. Some extensions of this result are also presented.

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Keywords: Conjugate functions, conjugate Fourier series, compact solenoidal groups
Article copyright: © Copyright 1983 American Mathematical Society