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

Abstract

This paper begins with a systematic study of noncommutative Fourier series on Z 2 \SE(2). Let µ be the finite SE(2)-invariant measure on the right coset space Z 2 \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 (Z 2 \SE(2), µ). We then investigate the presented theory for the case of a canonical basis originated from a fundamental domain of Z 2 in SE(2). The paper is concluded by some convolution results.