Let B be an almost-periodic (a.p.) function with mean value zero. 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 (Ω, 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 ωºεΩ such that b(ωº. 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 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) Haarmeasure on Ω, and let for some compact I⊂R and some ωεΩ. Here γ is Lebesgue Jo 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.

Almost-periodic functions with unbounded integral / R. Johnson. - In: PACIFIC JOURNAL OF MATHEMATICS. - ISSN 0030-8730. - STAMPA. - 87:(1980), pp. 347-362. [10.2140/pjm.1980.87.347]

Almost-periodic functions with unbounded integral

JOHNSON, RUSSELL ALLAN
1980

Abstract

Let B be an almost-periodic (a.p.) function with mean value zero. 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 (Ω, 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 ωºεΩ such that b(ωº. 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 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) Haarmeasure on Ω, and let for some compact I⊂R and some ωεΩ. Here γ is Lebesgue Jo 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.
1980
87
347
362
R. Johnson
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/396062
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