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^1/2 g and H^1/2 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∈W^2,2(S) with S=[-1/2, 1/2]×[-1/2, 1/2] and G(0), than 1/G∉L ²(S)