Abstract
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 ε 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 windowoγ ε 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 andoγ as knowns and unknowns, respectively. We briefly indicate when and how one can generate a Weyl-Heisenberg frame for the space\(\mathcal{P}_\mathcal{K} \) of K-periodic sequences, where K=LCM (N, M), by periodization of a Weyl-Heisenberg frame for ℓ2ℤ with shift parameters N, M−1.
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Janssen, A.J.E.M. From continuous to discrete Weyl-Heisenberg frames through sampling. The Journal of Fourier Analysis and Applications 3, 583–596 (1997). https://doi.org/10.1007/BF02648886
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DOI: https://doi.org/10.1007/BF02648886