Beginning with a mild
extension of a theorem of Littlewood, as generalised by Helgason and by
Grothendieck from the circle to a general compact Abelian group G, we derive some
properties of the Fourier series of continuous functions on G in relation to arbitrary
changes of sign of the coefficients. The main result of this latter type sharpens a
fact known for the circle by showing that a continuous function f on G
and a ±1-valued function ω on the character group X may be chosen so
that
belongs to no Orlicz space LΛ(G) for which limu→∞u−2Λ(u) = ∞. Similar results
are obtained which apply when f is assumed to be merely integrable: in
this case one can assert little more than that Tωf is a pseudomeasure on
G.
