Let G be a compact,
connected, Abelian group. Its dual, Γ, is discrete and can be ordered. Let Γ_{1} be a
semigroup which is a subset of the positive elements for some ordering, but which
contains the origin of Γ. Let H^{p}(Γ_{1}) be the subspace of L^{p}(G) consisting
of functions which have vanishing off Γ_{1}. The question that this paper is
concerned with is what conditions on a function in H^{p}(Γ_{1}) assure an innerouter
factorization.
An inner function is a function f ∈ H^{∞}(Γ_{1}) such that f = 1 a.e. (dx) on G. A
function f ∈ H^{p}(Γ_{1}) is said to be outer if
A function f ∈ H^{1}(Γ_{1}) is said to be in the class LRP(Γ_{1}) if log f∈ Γ_{1}(G) and
log f has Fourier coefficients equal to zero off Γ_{1} ∪−Γ_{1}. The main result of the
paper is that if Γ_{1} is the intersection of half planes and f ∈ H^{1}(Γ_{1}) with
∫
_{G}log f(x)dx > −∞ then f has an innerouter factorization if and only if log f is
in LRP(Γ_{1}).
