Two Theorems on Lattice Expansions

It is shown that there is a trade-off 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 [formula Omitted] are dual, then 1) at least one of H^1/2g and [formula Omitted] is not in [formula Omitted] at least one of Hg and [formula Omitted] is not in [formula Omitted]. Here, H is the operator-1/(4π²)d²/(dt²) + t². The first result is a generalization of a theorem first stated by Balian and independently by Low, which was recently rigorously proved by Coifman and Semmes; a new, much shorter proof was very recently given by Battle. Battle suggests a theorem of type (i), but our result is stronger in the sense that certain implicit assumptions made by Battle are removed. Result 2) is new and relies heavily on the fact that, when G E W^2,2(S) with [formula Omited] and G(0) = 0, then [formula Omitted]. The latter result was not known to us and may be of independent interest. © 1993 IEEE

Duke Scholars

Author Ingrid Daubechies Mathematics