Non-Abelian Fourier Series on $\mathbb{Z}^2\backslash SE(2)$

TL;DR

This paper develops a computational framework for non-Abelian Fourier series on the coset space of SE(2) modulo a lattice, enabling discrete sampling and analysis of functions and convolutions in this setting.

Contribution

It introduces a constructive method for sampling and computing non-Abelian Fourier coefficients on the coset space of SE(2), extending Fourier analysis tools to this non-commutative setting.

Findings

Provides a sampling scheme for Fourier matrix elements on SE(2)

Characterizes Fourier coefficients of square-integrable functions on the coset space

Discusses convolution properties in the non-Abelian Fourier context

Abstract

This paper discusses computational structure of coefficients of non-Abelian Fourier series on the right coset space $\mathbb{Z}^2\backslash SE(2)$ expressed in the trigonometric basis, where $SE(2)$ is the group of handedness preserving Euclidean isometries of the plane and $\mathbb{Z}^2$ denotes the discrete subgroup of translations of the orthogonal (square) lattice in $\mathbb{R}^2$. Assume that $\mu$ is the finite $SE(2)$-invariant measure on the right coset space $\mathbb{Z}^2\backslash SE(2)$, normalized with respect to Weil's formula. We present a constructive computational characterization including discrete sampling of non-Abelian Fourier matrix elements on $SE(2)$ for coefficients of $\mu$-square integrable functions on $\mathbb{Z}^2\backslash SE(2)$ with respect to the concrete trigonometric basis. The paper is concluded with discussion of the method for non-Abelian Fourier coefficients of convolutions on $\mathbb{Z}^2\backslash SE(2)$.