Validity ofWH-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 note it is shown that it actually depends critically on the timefrequency lattice used. Indeed, for any irrational > 0 we can construct a smooth g 2 L2(R) such that for any two rationals a > 0 and b > 0 the collection (gna;mb)n;m2Zof time-frequency translates of g has a finite frame upper bound, while for any > 0 and any rational c > 0 the collection (gnc ;m )n;m2Zhas 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 non-zero g 2 L2(R) which is sufficiently well-behaved, there exists ac > 0, bc > 0 such that (gna;mb)n;m2Z is a frame whenever 0 < a < ac, 0 < b < bc. We present two examples of a non-zero g 2 L2(R), bounded and supported by (0; 1), for which such numbers ac, bc do 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 a of the form l 3 k with l; k 2 N and all b, 0 < b < 1, while the frame lower bound is at least 1 for all a of the form (2m) 1 with m 2 N and all b, 0 < b < 1.