Three Problems on Trigonometric Sums

Let Λ ? R~n be a uniformly discrete set and let CΛ 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 ∈ CΛ we have f（x） = O(ωΛ（x）) as |x| →∞ and ωΛ（x） can be estimated（Theorem 4.1）. This line of research was proposed by Jean-Pierre Kahane in 1957.