From continuous to discrete Weyl-Heisenberg frames through sampling

In this article we consider the question when one can generate a Weyl- Heisenberg frame for l²(Z) with shift parameters N, M^-1 (integer N, M) by sampling a Weyl-Heisenberg frame for L²(R) with the same shift parameters at the integers. It is shown that this is possible when the window g e L² (R) generating the Weyl-Heisenberg frame satisfies an appropriate regularity condition at the integers. When, in addition, the Tolimieri-Orr condition A is satisfied, the minimum energy dual window °¿ e L²(R) can be sampled as well, and the two sampled windows continue to be related by duality and minimality. The results of this article also provide a rigorous basis for the engineering practice of computing dual functions by writing the Wexler-Raz biorthogonality condition in the time-domain as a collection of decoupled linear systems involving samples of g and °¿ as knowns and unknowns, respectively. We briefly indicate when and how one can generate a Weyl-Heisenberg frame for the space P¿ of K-periodic sequences, where K = LCM(N, M), by periodiization of a Weyl-Heisenberg frame for l²Z with shift parameters N, M^-1.