Sampling Theorems for Some Two-Step Nilpotent Lie Groups

Abstract

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, $\mathfrak{n=z}\oplus\mathfrak{b}\oplus\mathfrak{a}$, $\mathfrak{z}$ 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 $\mathfrak{z}\oplus\mathfrak{b}$ is a maximal commutative ideal of $\mathfrak{n},$ $\left[ \mathfrak{a,b}\right] \subseteq\mathfrak{z},$ and $\mathrm{det}\left([X_i,Y_j]\right)_{1\leq i,j\leq d}$ is a non-trivial homogeneous polynomial defined over the ideal $\left[ \mathfrak{n,n}\right] \subseteq\mathfrak{z}.$ We do not assume that $[\mathfrak{a},\mathfrak{a}]$ 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 $\Gamma$. The set $\Gamma$ is explicitly constructed by fixing a Jordan-H\"{o}lder basis for $\mathfrak{n}.$ We provide sufficient conditions for which a function $f$ is determined from its sampled values on $(f(\gamma))_{\gamma \in\Gamma}.$ We also provide an explicit formula for the corresponding sinc-type functions. Several examples are also computed in the paper.