Validity of WH-Frame Bound Conditions Depends on Lattice Parameters

AbstractIn 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∈ L2(R) such that for any two rationals a >0 and b >0 the collection (gna, mb)n, m∈ Zof time–frequency translates of ghas a finite frame upper bound, while for any β>0 and any rational c> 0 the collection (gncα, mβ)n, m∈ Zhas 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 ac>0, bc>0 such that (gn a, m b)n, m∈ Zis a frame whenever 0 < a < ac, 0 < b < bc. We present two examples of a nonzero g∈ L2(R), bounded and supported by (0, 1), for which such numbers ac, bcdo 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− kwith 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