Conjugate Fourier series on certain solenoids

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.