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, $\mathfrak{n}=\mathfrak{z}\oplus \mathfrak{b}\oplus \mathfrak{a}$, $\mathfrak{z}$ 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 $\mathfrak{z}\oplus \mathfrak{b}$ is a maximal commutative ideal of $\mathfrak{n},$ $\left[\mathfrak{a},\mathfrak{b}\right]\subseteq \mathfrak{z},$ and $\det {\left([X_i,Y_j]\right)}_{1\le i,j\le d}$ is a non-trivial homogeneous polynomial defined over the ideal $\left[\mathfrak{n},\mathfrak{n}\right]\subseteq \mathfrak{z}.$ 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-Hölder 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.

References in corpus (2)

• Shannon Multiresolution Analysis on the Heisenberg Group
• Shannon-Like Wavelet Frames on a Class of Nilpotent Lie Groups