Frames of Translates for Number-Theoretic Groups

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 $${\mathbb {R}}^d$$ .