Variants of Müntz–Szàsz Analogs for Euclidean Spin Groups

Given a strictly increasing sequence of positive integers $(n_k)_k$, the Müntz–Szàsz theorem for completeness of the monomials $\{x^{n_k}\}$ in $L^2([0,1])$ can be extended to Euclidean spin groups which are the universal coverings of Euclidean motion groups $SO(n) \ltimes \mathbb{R}^n$. Towards such an objective, we rephrase the completeness condition in terms of an integral against the monomials $\{x^{n_k}\}$ of the coordinates functions associated to the Garding vectors of $G$, whose Fourier transform are shown to admit an analytic continuation to the whole complex plane with an exponential domination. This allows us to formulate and prove several variants of Müntz–Szàsz's theorem in these settings. These upshots are proved using the Plancherel theory related to the group Fourier transform.