Vol. 87, No. 2, 1980

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Almost-periodic functions with unbounded integral

Russell Allan Johnson

Vol. 87 (1980), No. 2, 347–362

Let B be an almost-periodic (a.p.) function with mean value zero. Let G(t) = 0tB(s)ds. The well-known theorem of Bohr states that G(t) is uniformly bounded iff G(t) is a.p. This theorem may be reformulated in the following way. Let Ω be the hull of B, and let ,R) be the flow on Ω defined by translation. Since B is a.p., Ω is a compact abelian topological group. There is a continuous b : Ω R and an ω0 Ω such that b(ω0 t) = B(t). I.e., b “extends B to Ω”. Then Bohr’s theorem is equivalent to the following: G(t) is bounded iff there is a continuous r : Ω R such that r(ω t) r(ω) = 0tb(ω s)ds (ω Ω,t R).

In this paper, we consider the case when G(t) is unbounded. Two results are obtained. The first is a generalization of Bohr’s theorem: let μ be (normalized) Haar measure on Ω, and let gω(t) = 0tb(ω s)ds (ω Ω,t R); then lim-n→∞12{t [n,n]|gω(t) I} > 0 for some compact I R and some ω Ω iff there exists a μ-measurable r : Ω R such that r(ω t) r(ω) = 0tb(ω s)ds (ω Ω,t R). Here γ is Lebesgue measure on R. Thus, r exists if some gω(t) is not too badly unbounded. This theorem is stated for the class of “minimal” functions (see below), which includes the a.p. ones.

Mathematical Subject Classification 2000
Primary: 42A75
Secondary: 28D99
Received: 15 February 1978
Revised: 28 March 1979
Published: 1 April 1980
Russell Allan Johnson