Zak Transform Associated with the Weyl Transform and the System of Twisted Translates on R2n.

We introduce the Zak transform on L 2 (R 2 n) associated with the Weyl transform. By making use of this transform, we define a bracket map and prove that the system of twisted translates { T (k , l) t ϕ : k , l ∈ Z n } is a frame sequence iff 0 < A ≤ ϕ , ϕ (ξ , ξ ′) ≤ B < ∞ , for a.e (ξ , ξ ′) ∈ Ω ϕ , where Ω ϕ = { (ξ , ξ ′) ∈ T n × T n : ϕ , ϕ (ξ , ξ ′) ≠ 0 } . We also prove a similar result for the system { T (k , l) t ϕ : k , l ∈ Z n } to be a Riesz sequence. For a given function belonging to the principal twisted shift-invariant space V t (ϕ) , we find a necessary and sufficient condition for the existence of a canonical biorthogonal function. Further, we obtain a characterization for the system { T (k , l) t ϕ : k , l ∈ Z } to be a Schauder basis for V t (ϕ) in terms of a Muckenhoupt A 2 weight function. [ABSTRACT FROM AUTHOR]