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On characterizations of Weyl-Heisenberg frame sets. (English) Zbl 1191.42016

Summary: A Weyl-Heisenberg frame for \(L^2(\mathbb R)\) is a frame consisting of the set of translates and modulates of a fixed function in \(L^2(\mathbb R)\) i.e., \( \{E_{mb}T_{na}g\}_{m,n\in \mathbb Z}\) with \(a,b >0\) and \(g\in L^2(\mathbb R)\) A fundamental question is to explicitly represent the families \((g,a,b)\) such that \(\{E_{mb}T_{na}g\}_{m,n\in \mathbb Z}\) is a frame for \(L^2(\mathbb R)\) In this paper, we investigate the difficult problem at the rationally oversampled case. Let \(E = [0,1)+\{n_1,\dots ,n_k\}\) where \(\{n_1,\dots ,n_k\}\subset \mathbb Z\) We show that \(E\) is a Weyl-Heisenberg frame set for \((p/q,1)\) is equivalent to classifying the integer sets \(\{n_1,\dots ,n_k\}\) such that \(p(z) ) \sum _{j=1}^k z^{n_j}\) does not have any zeros on the unit circle in the plane. To show our results the technique in the Zak transform has been used.

MSC:

42C15 General harmonic expansions, frames
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