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 α>0 we can construct a smooth g∈ L 2( 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 β>0 and any rational c> 0 the collection ( g ncα, mβ ) 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∈ L 2( 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∈ L 2( R ), bounded and supported by (0, 1), for which such numbers a c , b c 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 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) − 1with m∈ N and all b, 0 < b <1.
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