Frames of translates for number-theoretic groups

TL;DR

This paper characterizes frames of translates in L^2(G) for number-theoretic groups, including local fields, using spectral symbols and a novel translation concept, extending previous results from Euclidean spaces.

Contribution

It introduces a new translation notion and characterizes frames of translates in L^2(G) for number-theoretic groups, generalizing prior Euclidean space results.

Findings

Characterization of frames via spectral symbols

Introduction of a new translation concept

Extension to local fields and other 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 Shidong 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.