Density Results for Frames of Exponentials

Abstract

For a separated sequence Λ={λ _k } _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 $\{ e^{i\lambda _k x} \} _{k \in \mathbb{Z}:}$ in fact, $\{ e^{i\lambda _k x} \} _{k \in \mathbb{Z}}$ 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 $\{ e^{i\lambda _k x} \} _{k \in \mathbb{Z}}$ has no frame property in L ²(−ν, ν) for any ν ∈ (π D ⁻(Λ), π D ⁺(Λ)).