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

Let ϕ be an arbitrary generalized Gaussian (squeezed coherent state), Λ_{αβ}=(α_1 Z \times\cdot\cdot\cdot\times α_{n}\mathbb{Z)\times}(β_{1}\mathbb{Z}\times\cdot\cdot\cdot \timesβ_{n}\mathbb{Z)}$ a rectangular lattice. We show that there exists a positive definite symplectic matrix M (depending on ϕ) such that the multivariate Weyl--Heisenberg system G(ϕ,MΛ_{αβ}) is a frame. In a forthcoming Note we will prove a converse to this result.

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