Construction of Multivariate Gaussian Weyl--Heisenberg Frames, (I)

Let {\phi} be an arbitrary generalized Gaussian (squeezed coherent state), {\Lambda}_{{\alpha}{\beta}}=({\alpha}_1 Z \times\cdot\cdot\cdot\times \alpha_{n}\mathbb{Z)\times}(\beta_{1}\mathbb{Z}\times\cdot\cdot\cdot \times\beta_{n}\mathbb{Z)}$ a rectangular lattice. We show that there exists a positive definite symplectic matrix M (depending on {\phi}) such that the multivariate Weyl--Heisenberg system G({\phi},M{\Lambda}_{{\alpha}{\beta}}) is a frame. In a forthcoming Note we will prove a converse to this result.