Conjugate Fourier series on certain solenoids
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- by Edwin Hewitt and Gunter Ritter PDF
- Trans. Amer. Math. Soc. 276 (1983), 817-840 Request permission
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 $(\tilde {f})^{\hat {}}(\alpha )$ 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.References
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Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 276 (1983), 817-840
- MSC: Primary 43A70; Secondary 42A50
- DOI: https://doi.org/10.1090/S0002-9947-1983-0688979-9
- MathSciNet review: 688979