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Document type:
Buchbeitrag 
Author(s):
Casazza, P. G., Christensen, O., Li, S. and Lindner, A. 
Title:
Density results for frames of exponentials 
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 {ekx}kεZ: in fact, {ekx}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 {ekx} has no frame property in L2(-γ,γ) for any γε ]πD-(Λ),πD+(Λ)[. 
Book title:
Heil, C.: Harmonic Analysis and Applications 
Book subtitle:
In Honor of John J. Benedetto 
Publisher:
Birkhäuser 
Year:
2006 
Reviewed:
ja 
Language:
en 
Semester:
SS 06 
Format:
Text