The canonical group of transformations of a Weyl-Heisenberg frame; applications to Gaussian and Hermitian frames

We show, using a simple trick due to E. Fermi and rediscovered by Benenti and Strini, that to every Weyl-Heisenberg frame is associated in a canonical way a one-parameter group of symplectic transformations acting on the lattice of that frame without affecting the window. We study in detail the case of multidimensional Gaussian and Hermitian frames, and show that our method allows to create infinitely many new frames with same window staring from a single frame.