Regular Sampling on Metabelian Nilpotent Lie Groups

Let $N$ be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra $\mathfrak{n}$ having rational structure constants. We assume that $N=P\rtimes M,$ $M$ is commutative, and for all $λ\in \mathfrak{n}^{\ast}$ in general position the subalgebra $\mathfrak{p}=\log(P)$ is a polarization ideal subordinated to $λ$ ($\mathfrak{p}$ is a maximal ideal satisfying $[\mathfrak{p},\mathfrak{p}]\subseteq\kerλ$ for all $λ$ in general position and $\mathfrak{p}$ is necessarily commutative.) Under these assumptions, we prove that there exists a discrete uniform subgroup $Γ\subset N$ such that $L^{2}(N)$ admits band-limited spaces with respect to the group Fourier transform which are sampling spaces with respect to $Γ.$ We also provide explicit sufficient conditions which are easily checked for the existence of sampling spaces. Sufficient conditions for sampling spaces which enjoy the interpolation property are also given. Our result bears a striking resemblance with the well-known Whittaker-Kotel'nikov-Shannon sampling theorem.

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