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Two theorems on lattice expansions

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2 Author(s)
Daubechies, I. ; AT&T Bell Labs., Murray Hill, NJ, USA ; Janssen, A.

It is shown that there is a tradeoff between the smoothness and decay properties of the dual functions, occurring in the lattice expansion problem. More precisely, it is shown that if g and g¯ are dual, then (1) at least one of H1/2 g and H1/2 g¯ is n in L2(R), and (2) at least one of Hg and g ¯ is not in L2(R). Here, H is the operator -1/(4π2)d2/(dt2 )+t2. The first result is a generalization of a theorem first stated by R.C. Balian (1987). The second result is new and relies heavily on the fact that, when G∈W2,2(S) with S=[-1/2, 1/2]×[-1/2, 1/2] and G(0), than 1/GL 2(S)

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Information Theory, IEEE Transactions on  (Volume:39 ,  Issue: 1 )