Validity of WH-frame bound conditions depends on lattice parameters (Q1969041)

It is well known that when \(g \in L^2(\mathbb R)\) is ``sufficiently well-behaved'', then there exist constants \(a_c, b_c >0\) such that \(\{e^{2\pi mbx}(g(x-na)\}_{m,n\in \mathbb Z}\) is a frame for \(L^2(\mathbb R)\) for all \(a\in ]0,a_c], b\in ]0,b_c]\). The present paper shows by examples what can happen when \(g\) is not ``sufficiently well-behaved''. An example of a function \(g\neq 0\) for which no such numbers \(a_c, b_c\) exist is given. Also, an (amazing!) example is given, where (i) \(\{e^{2\pi mbx}(g(x-na)\}_{m,n\in \mathbb Z}\) is a frame for all \(a= \frac 1{2m}, \;m\in {\mathbb N}\) and \(b\in ]0,1[\), and (ii) \(\{e^{2\pi mbx}(g(x-na)\}_{m,n\in \mathbb Z}\) is never a frame when \(a= \frac{l}{3^k}, \;k,l \in {\mathbb N}\) and \(b\in ]0,1[\).