Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 1

Special Issue in Honor of Bernie Shiffman

Guest Editors: Yuan Yuan, Christopher Sogge, and Steven Morris Zelditch

Abstract noncommutative Fourier series on $\Gamma \setminus SE(2)$

Pages: 71 – 100

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n1.a3


Arash Ghaani Farashahi (Department of Mechanical Engineering, National University of Singapore; and Department of Pure Mathematics, School of Mathematics, University of Leeds, United Kingdom)

Gregory Chirikjian (Department of Mechanical Engineering, National University of Singapore)


This paper begins with a systematic study of abstract noncommutative Fourier series on $\Gamma \setminus SE(2)$, where $\Gamma$ is a discrete co-compact subgroup of $SE(2)$, the group of all handedness-preserving isometries of the Euclidean plane. Let $\mu$ be the finite $SE(2)$-invariant measure on the right coset space $\Gamma \setminus SE(2)$, 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 $L^2 (\Gamma \setminus SE(2), \mu)$. The paper concludes with some convolution results.


special Euclidean group, non-commutative Fourier series, coset space, discrete subgroup, crystallographic subgroup

2010 Mathematics Subject Classification

Primary 43A10, 43A15, 43A20, 43A30, 43A85. Secondary 20H15, 68T40, 74E15, 82D25.

Received 15 June 2020

Accepted 16 November 2020

Published 10 February 2022