Abstract
Frames of translates of \(f \in L^2(G)\) are characterized in terms of the zero-set of the so-called spectral symbol of f in the setting of a locally compact abelian group G having a compact open subgroup H. We refer to such a G as a number-theoretic group. This characterization was first proved in 1992 by Li and one of the authors for \(L^2({\mathbb {R}}^d)\) with the same formal statement of the characterization. For number-theoretic groups, and these include local fields, the strategy of proof is necessarily entirely different, and it requires a new notion of translation that reduces to the usual definition in \({\mathbb {R}}^d\).
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The first-named author gratefully acknowledges the support of ARO Grant W911NF-17-1-0014 and NSF-ATD Grant DMS-1738003. The second named author gratefully acknowledges the support of NSF Grant DMS-1501766. The authors appreciate helpful comments by Carlos Cabrelli, Karlheinz Gröchenig, Eugenio Hernández, and Victoria Paternostro. Finally, the authors are grateful for the constructive and thorough reviews by the two anonymous referees. All of their suggestions have been incorporated into this final version.
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Benedetto, J.J., Benedetto, R.L. Frames of Translates for Number-Theoretic Groups. J Geom Anal 30, 4126–4149 (2020). https://doi.org/10.1007/s12220-019-00234-y
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DOI: https://doi.org/10.1007/s12220-019-00234-y