On discrete groups of Euclidean isometries: representation theory, harmonic analysis and splitting properties

TL;DR

This paper investigates the structure and harmonic analysis of discrete Euclidean isometry groups, providing a dual space description, Fourier analysis techniques, and a splitting theorem, advancing understanding of their algebraic and analytical properties.

Contribution

It introduces a comprehensive description of the dual space, develops Fourier analysis methods for periodic functions, and proves a splitting theorem for discrete Euclidean isometry groups.

Findings

Dual space of discrete Euclidean isometry groups characterized

Fourier analysis methods for periodic mappings developed

A Schur-Zassenhaus type splitting theorem proved

Abstract

We study structural properties and the harmonic analysis of discrete subgroups of the Euclidean group. In particular, we 1. obtain an efficient description of their dual space, 2. develop Fourier analysis methods for periodic mappings on them, and 3. prove a Schur-Zassenhaus type splitting result.