Limits of integrals involving almost periodic functions

Abstract

Let Sp ⊂ R⁺ be a discrete countable set, let {α_λ}_λ∈Sp be a sequence in l¹(Sp) and f(x) ≔ ∑_λ∈Spα_λsin(λx). f is an almost periodic odd function with {λ: ±λ ∈ Sp} as spectrum. We give some conditions about the set S so that $\int _1^{+\infty}\ f(x){\rm sin}(Rx){dx\over x}\rightarrow 0$ whenever R → +∞, R ∈ S.