Non-Abelian Fourier Analysis on Γ SE(d)

This paper presents a systematic study for the general theory of non-Abelian Fourier series of integrable functions on the homogeneous space Γ SE(d), where $SE(d)$ is the special Euclidean group in dimension d, and Γ is a discrete and co-compact subgroup of $SE(d)$. Suppose that μ is the finite $SE(d)$-invariant measure on the right coset space Γ SE(d), normalized with respect to Weil's formula. The analytic aspects of the proposed method works for any given orthonormal basis of the Hilbert function space L²(Γ SE(d),μ). The paper is concluded with some convolution and Plancherel formulas.