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This paper discusses computational structure of coefficients of non-AbelianFourier series on the right coset space $\mathbb{Z}^2\backslash SE(2)$expressed in the trigonometric basis, where $SE(2)$ is the group of handednesspreserving Euclidean isometries of the plane and $\mathbb{Z}^2$ denotes thediscrete subgroup of translations of the orthogonal (square) lattice in$\mathbb{R}^2$. Assume that $\mu$ is the finite $SE(2)$-invariant measure onthe right coset space $\mathbb{Z}^2\backslash SE(2)$, normalized with respectto Weil's formula. We present a constructive computational characterizationincluding discrete sampling of non-Abelian Fourier matrix elements on $SE(2)$for coefficients of $\mu$-square integrable functions on$\mathbb{Z}^2\backslash SE(2)$ with respect to the concrete trigonometricbasis. The paper is concluded with discussion of the method for non-AbelianFourier coefficients of convolutions on $\mathbb{Z}^2\backslash SE(2)$.

Non-Abelian Fourier Series on $\mathbb{Z}^2\backslash SE(2)$