Two theorems on lattice expansions

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 H/sup 1/2/ g and H/sup 1/2/ g is n in L/sup 2/(R), and (2) at least one of Hg and g is not in L/sup 2/(R). Here, H is the operator -1/(4 pi /sup 2/)d/sup 2//(dt/sup 2/)+t/sup 2/. 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 in W/sup 2,2/(S) with S=(-1/2, 1/2)*(-1/2, 1/2) and G(0), than 1/G not in L/sup 2/(S).<
>