Pure and Applied Mathematics Quarterly
Volume 18 (2022)
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
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