Noncommutative Fourier Series on $\mathbb{Z}^2\backslash SE(2)$

Abstract

This paper begins with a systematic study of noncommutative Fourier series on $\mathbb{Z}^2\backslash SE(2)$. Let $\mu$ be the finite $SE(2)$-invariant measure on the right coset space $\mathbb{Z}^2\backslash SE(2)$, normalized with respect to Weil's formula. The analytic aspects of the proposed method works for any given (discrete) basis of the Hilbert function space $L^2(\mathbb{Z}^2\backslash SE(2),\mu)$. We then investigate the presented theory for the case of a canonical basis originated from a fundamental domain of $\mathbb{Z}^2$ in $SE(2)$. The paper is concluded by some convolution results