## Abstract.

Let *N* be a simply connected, connected
nilpotent Lie group with the following assumptions. Its Lie algebra $\U0001d52b$ is an *n*-dimensional vector space over the reals. Moreover, $\U0001d52b=\U0001d537\oplus \U0001d51f\oplus \U0001d51e$, $\U0001d537$
is the center of $\U0001d52b$,
$\U0001d537=\mathbb{R}{Z}_{n-2d}\oplus \mathbb{R}{Z}_{n-2d-1}\oplus \cdots \oplus \mathbb{R}{Z}_{1}$, $\U0001d51f=\mathbb{R}{Y}_{d}\oplus \mathbb{R}{Y}_{d-1}\oplus \cdots \oplus \mathbb{R}{Y}_{1}$,
$\U0001d51e=\mathbb{R}{X}_{d}\oplus \mathbb{R}{X}_{d-1}\oplus \cdots \oplus \mathbb{R}{X}_{1}$. Next, assume $\U0001d537\oplus \U0001d51f$ is a maximal commutative ideal of
$\U0001d52b$, $[\U0001d51e,\U0001d51f]\subseteq \U0001d537$, and $\mathrm{det}{\left([{X}_{i},{Y}_{j}]\right)}_{1\le i,j\le d}$ is a non-trivial
homogeneous polynomial defined over the ideal $[\U0001d52b,\U0001d52b]\subseteq \U0001d537$. We do not assume that $[\U0001d51e,\U0001d51e]$ is generally trivial. We obtain some precise description of band-limited spaces which are sampling subspaces of ${L}^{2}\left(N\right)$ with respect to some discrete set $\Gamma $. The set $\Gamma $ is explicitly constructed by fixing a strong Malcev basis for $\U0001d52b$. We provide sufficient conditions for which a function *f* is determined from its sampled values on ${\left(f\left(\gamma \right)\right)}_{\gamma \in \Gamma}$. We also provide an explicit formula for the corresponding sinc-type functions. Several examples are also computed in the paper.

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