  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 $\lambda\in
\mathfrak{n}^{\ast}$ in general position the subalgebra $\mathfrak{p}=\log(P)$
is a polarization ideal subordinated to $\lambda$ ($\mathfrak{p}$ is a maximal
ideal satisfying $[\mathfrak{p},\mathfrak{p}]\subseteq\ker\lambda$ for all
$\lambda$ in general position and $\mathfrak{p}$ is necessarily commutative.)
Under these assumptions, we prove that there exists a discrete uniform subgroup
$\Gamma\subset N$ such that $L^{2}(N)$ admits band-limited spaces with respect
to the group Fourier transform which are sampling spaces with respect to
$\Gamma.$ 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.
