We consider an arbitrary noncyclic subgroup of the additive group
Q
{\mathbf {Q}}
of rational numbers, denoted by
Q
a
{{\mathbf {Q}}_{\mathbf {a}}}
, and its compact character group
Σ
a
{\Sigma _{\mathbf {a}}}
. For
1
>
p
>
∞
1 > p > \infty
, an abstract form of Marcel Riesz’s theorem on conjugate series is known. For
f
f
in
L
p
(
Σ
a
)
{\mathfrak {L}_p}({\Sigma _{\mathbf {a}}})
, there is a function
f
~
\tilde {f}
in
L
p
(
Σ
a
)
{\mathfrak {L}_p}({\Sigma _{\mathbf {a}}})
whose Fourier transform
(
f
~
)
^
(
α
)
(\tilde {f})^{\hat {}}(\alpha )
at
α
\alpha
in
Q
a
{{\mathbf {Q}}_{\mathbf {a}}}
is
−
i
sgn
α
f
^
(
α
)
- i\,\operatorname {sgn}\,\alpha \hat {f}(\alpha )
. We show in this paper how to construct
f
~
\tilde {f}
explicitly as a pointwise limit almost everywhere on
Σ
a
{\Sigma _{\mathbf {a}}}
of certain harmonic functions, as was done by Riesz for the circle group. Some extensions of this result are also presented.