Non-Abelian Fourier Analysis on SE (d)

This paper presents a systematic study for the general theory of non-Abelian Fourier series of integrable functions on the homogeneous space SE (d), where SE (d) is the special Euclidean group in dimension d, and is a discrete and co-compact subgroup of SE (d). Suppose that is the finite SE (d) -invariant measure on the right coset space 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 (SE (d), ). The paper is concluded with some convolution and Plancherel formulas.