Validity of WH-frame bound conditions depends on lattice parameters

In the study of Weyl-Heisenberg frames the assumption of having a finite fr ame upper bound appears recurrently. In this article it is shown that it ac tually depends critically on the time-frequency lattice used. Indeed, for a ny irrational alpha > 0 we can construct a smooth g is an element of L-2(R) such that for any two rationals a > 0 and b > 0 the collection (g(na,mb))( n),(m is an element of Z) of time-frequency translates of g has a finite fr ame upper bound, while for any beta > 0 and any rational c > 0 the collecti on (g(nc alpha,m beta))(n,m is an element of Z) has no such bound. It follo ws from a theorem of I. Daubechies, as well as from the general atomic theo ry developed by Feichtinger and Grochenig, that for any nonzero g is an ele ment of L-2(R) which is sufficiently well behaved, there exist a(c) > 0, b( c) > 0 such that (g(na,mb))(n,m is an element of Z) is a frame whenever 0 < a < a(c), 0 < b < b(c). We present two examples of a nonzero g is an eleme nt of 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 equ als 0 for all a > 0, b > 0, b < 1. In the second example, the frame lower b ound equals 0 for all a of the form I 3(-k) with I, k is an element of N an d all b, 0 < b < 1, while the frame lower bound is at least 1 for all a of the form (2,m)(-1) with,n is an element of N and all b, 0 < b < 1. (C) 2000 Academic Press.