Li, Hai-Xiong; Deng, Guo-Tai On characterizations of Weyl-Heisenberg frame sets. (English) Zbl 1191.42016 Curr. Dev. Theory Appl. Wavelets 3, No. 3, 219-232 (2009). 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 Keywords:WH frame; Zak transform; rational oversampling; characteristic function PDFBibTeX XMLCite \textit{H.-X. Li} and \textit{G.-T. Deng}, Curr. Dev. Theory Appl. Wavelets 3, No. 3, 219--232 (2009; Zbl 1191.42016) Full Text: Link