Validity of WH-frame bound conditions depends on Lattice parameters

In the study of Weyl-Heisenberg frames the assumption of having a finite frame upper bound appears recurrently. In this article it is shown that it actually depends critically on the time-frequency lattice used. Indeed, for any irrational a>0 we can construct a smooth g¿ L²(R) such that for any two rationals a >0 and b >0 the collection (g_{na, mb})_{n, m¿ Z}of time-frequency translates of ghas a finite frame upper bound, while for any ß>0 and any rational c> 0 the collection (g_{nca, mß})_{n, m¿ Z}has no such bound. It follows from a theorem of I. Daubechies, as well as from the general atomic theory developed by Feichtinger and Gröchenig, that for any nonzero g¿ L²(R) which is sufficiently well behaved, there exist a_c>0, b_c>0 such that (g_{n a, m b})_{n, m¿ Z}is a frame whenever 0 < a < a_c, 0 < b < b_c. We present two examples of a nonzero g¿ L²(R), bounded and supported by (0, 1), for which such numbers a_c, b_cdo not exist. In the first one of these examples, the frame bound equals 0 for all a >0, b >0, b <1. In the second example, the frame lower bound equals 0 for all aof the form l· 3^{- k}with l, k¿ N and all b, 0 < b <1, while the frame lower bound is at least 1 for all aof the form (2 m)^{- 1}with m¿ N and all b, 0 < b <1.