Density Results for Frames of Exponentials

Abstract

For a separated sequence Λ={λk}k∈z of real numbers there is a close link between the lower and upper densities D −(Λ), D +(Λ) and the frame properties of the exponentials (formula presented) in fact, (formula presented) is a frame for its closed linear span in L 2(−ν, ν) for any ν ∈ (0, πD -(Λ)) ∪ (πD +(Λ),∞). We consider a classical example presented already by Levinson [11] with D -(Λ) = D +(Λ) = 1; in this case, the frame property is guaranteed for all ν ∈ (0; ∞) ∖ {π}. We prove that the frame property actually breaks down for ν = π. Motivated by this example, it is natural to ask whether the frame property can break down on an interval if D −(Λ) ≠ D +(Λ). The answer is yes: We present an example of a family Λ with D −(Λ) ≠ D +(Λ) for which (formula presented) has no frame property in L 2(−ν, ν) for any ν ∈ (π D −(Λ), π D +(Λ)).