Casazza, P. G., Christensen, O., Li, S. and Lindner, A.
Pages contribution:
359-369
Abstract:
For a separated sequence Λ={λk}kεZ 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 {eiλkx}kεZ: in fact, {eiλkx}kεZ is a frame for its closed linear
span in L2(-γ,γ) for any γε ]0,πD-(Λ)[ ∪ ]πD+(Λ),∞[. We consider a classical
example presented already by Levinson [10] with D-(Λ) = D+(Λ) = 1; in this case,
the frame property is guaranteed for all
γε]0,∞[. We prove that the frame
property actually breaks down for
γ=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 {eiλkx} has no frame property in L2(-γ,γ) for any
γε ]πD-(Λ),πD+(Λ)[.