Sampling Theorems for Some Two-Step Nilpotent Lie Groups

Let $N$ be a simply connected, connected nilpotent Lie group with the following assumptions. Its Lie Lie algebra $\mathfrak{n}$ is an $n$-dimensional vector space over the reals. Moreover, $\mathfrak{n}=\mathfrak{z}\oplus \mathfrak{b}\oplus \mathfrak{a}$, $\mathfrak{z}$ is the center of $\mathfrak{n}$, $\mathfrak{z} =$\mathbb{R}$Z_{n-2d}\oplus$\mathbb{R}$Z_{n-2d-1}\oplus\cdots\oplus $\mathbb{R}$Z_{1}, \mathfrak{b} =$\mathbb{R}$Y_{d}\oplus$\mathbb{R}$ Y_{d-1}\oplus\cdots\oplus$\mathbb{R}$Y_{1}, \mathfrak{a} =$\mathbb{R}$X_{d}\oplus$\mathbb{R}$X_{d-1}\oplus\cdots\oplus $\mathbb{R}$X_{1}.$Next,assume$\mathfrak{z}\oplus\mathfrak{b}$isamaximalcommutativeidealof$\mathfrak{n},$ $\left[\mathfrak{a},\mathfrak{b}\right]\subseteq \mathfrak{z},$ and $\mathrm{d}\mathrm{e}\mathrm{t}{([X_i,Y_j])}_{1\le i,j\le d}$ is a non-trivial homogeneous polynomial defined over the ideal $\left[ \mathfrak{n,n}\right] \subseteq\mathfrak{z}.$Wedonotassumethat$[\mathfrak{a},\mathfrak{a}]$ is generally trivial. We obtain some precise description of band-limited spaces which are sampling subspaces of $L^2(N)$ with respect to some discrete set $\mathrm{\Gamma }$. The set $\mathrm{\Gamma }$ is explicitly constructed by fixing a Jordan-H"{o}lder basis for $\mathfrak{n}\mathrm{.}$ We provide sufficient conditions for which a function $f$ is determined from its sampled values on $(f(\gamma ){)}_{\gamma \in \mathrm{\Gamma }}\mathrm{.}$ We also provide an explicit formula for the corresponding sinc-type functions. Several examples are also computed in the paper.