Frames of Translates for Number-Theoretic Groups
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Frames of Translates for Number-Theoretic Groups John J. Benedetto1
· Robert L. Benedetto2
Received: 23 May 2018 © Mathematica Josephina, Inc. 2019
Abstract Frames of translates of f ∈ 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 (Rd ) 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 Rd . Keywords Compact open subgroups · Frames of translates · Local fields · Spectral symbol Mathematics Subject Classification 43A25
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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John J. Benedetto [email protected] http://www.math.umd.edu/~jjb Robert L. Benedetto [email protected] https://rlbenedetto.people.amherst.edu/
1
Department of Mathematics, Norbert Wiener Center, University of Maryland, College Park, MD 20742, USA
2
Department of Mathematics and Statistics, Amherst College, Amherst, MA, USA
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J. J. Benedetto, R. L. Benedetto
1 Introduction 1.1 Background Time frequency analysis, wavelet theory, the theory of frames, sampling theory, and shift invariant spaces have not only burgeoned, but have also uncovered a host specific, tantalizing problem areas. One of these is the frame theoretic characterization of a closed span of translations. This is our bailiwick here, and it has become a topic with great generalization, applicability, intricacy, and abstraction, and with a large number of contributors, see, e.g., [1,2,9,11–13,16,17,22] and the references therein. We shall focus on the setting of what we call number-theoretic LCAGs, and by which we mean locally compact abelian groups (LCAGs) G with a compact open subgroup H . For a given function f on G, we shall solve the particular problem in this setting of characterizing when the closed span of translates of f is a frame. The characterization is in terms of the zero-set of a natural spectral symbol, see Theorem 4.5 for the solution. This closed span of translates problem is also addressed in the aforementioned references but not for number-theoretic groups. The Euclidean version, going back to 1992, is restated in Theorem 1.2. The strategy for its proof is natural, whereas the proof of Theorem 4.5 requires a new idea that we explain. In Se
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