Vol. 69, No. 2, 1977

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Inner-outer factorization of functions whose Fourier series vanish off a semigroup

Howard Lewis Penn

Vol. 69 (1977), No. 2, 501–504

Let G be a compact, connected, Abelian group. Its dual, Γ, is discrete and can be ordered. Let Γ1 be a semigroup which is a subset of the positive elements for some ordering, but which contains the origin of Γ. Let Hp1) be the subspace of Lp(G) consisting of functions which have vanishing off Γ1. The question that this paper is concerned with is what conditions on a function in Hp1) assure an inner-outer factorization.

An inner function is a function f H1) such that |f| = 1 a.e. (dx) on G. A function f Hp1) is said to be outer if

∫               ∫
log|f(x )| = log|  f(x )dx| > − ∞.
G               G

A function f H11) is said to be in the class LRP1) if log |f|∈ Γ1(G) and log |f| has Fourier coefficients equal to zero off Γ1 ∪−Γ1. The main result of the paper is that if Γ1 is the intersection of half planes and f H11) with Glog |f(x)|dx > −∞ then f has an inner-outer factorization if and only if log |f| is in LRP1).

Mathematical Subject Classification 2000
Primary: 43A70
Secondary: 46J10
Received: 19 November 1974
Revised: 8 November 1976
Published: 1 April 1977
Howard Lewis Penn