Noncommutative Fourier Series on Z2\SE(2)\mathbb{Z}^2\backslash SE(2)

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

Operator-Valued Continuous Gabor Transforms over Non-unimodular Locally Compact Groups

In this article, we present the abstract harmonic analysis aspects of the operator-valued continuous Gabor transform (CGT) on second countable, non-unimodular, and type I locally compact groups. We show that the operator-valued continuous Gabor transform CGT satisfies a Plancherel formula and an inversion formula. As an example, we study these results on the continuous affine group