Frames of Translates for Number-Theoretic Groups

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 19...

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Bibliographic Details
Published in:The Journal of geometric analysis 2020-12, Vol.30 (4), p.4126-4149
Main Authors: Benedetto, John J., Benedetto, Robert L.
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Group theory
Mathematics
Mathematics and Statistics
Subgroups
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Summary: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 ( 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 R d .
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-019-00234-y