Three Problems on Trigonometric Sums

Let Λ ⊂ ℝn be a uniformly discrete set and let C_Λ be the vector space consisting of all mean periodic functions whose spectrum is simple and contained in Λ. If Λ is a gentle set then for every f ∈ C_Λ we have f(x) = O(ω_Λ(x)) as |x| → ∞ and ω_Λ(x) can be estimated (Theorem 4.1). This line of research was proposed by Jean-Pierre Kahane in 1957.