The structure of Gabor systems and the theory of τ-pseudo-differential operators

We consider pseudo-differential operators and Gabor frames on locally compact commutative Hausdorff groups. Let [Formula: see text] be an open subgroup of a locally compact commutative Hausdorff group [Formula: see text] and [Formula: see text] be a topological isomorphism, then we define the cross [Formula: see text]-ambiguity function [Formula: see text] and cross [Formula: see text]-Wigner function [Formula: see text] and a pseudo-differentiable operator [Formula: see text] with the symbol [Formula: see text]. Employing properties of [Formula: see text], we establish characteristics of pseudo-differentiable operators. Also, assume that [Formula: see text] and [Formula: see text], then the coefficient operator [Formula: see text] uniquely, continuously extends to the linear operator [Formula: see text], where [Formula: see text] is the restriction of [Formula: see text] to [Formula: see text]; and similarly, the adjoint operator [Formula: see text] uniquely, continuously extends to the linear operator [Formula: see text], where [Formula: see text] is the restriction of [Formula: see text] to [Formula: see text].