Logic Density in Gabor Frame Structures

The Gabor system G(g,1+ε,ε-1)={e^2iπ(ε-1)nt g(t-(1+ε)m):m,n∈Z}is examined with respect to its frame property, uunder reasonable oversampling conditions, that is, when (ε+1,ε-1∈Q). An appropriate "rational" counterpart of the Ron-Shen Gramian is developed, establishing that for every peculiar pane function (g), The structure: G(g,ε+1,ε-1) fails to produce a frame if ε^2=(2n-1)/(n-1). A particular focus is placed on the initial Hermite function, h_1=te^(-πt^2 ).