Sampling and Interpolation on Some Non-commutative Nilpotent Lie Groups

TL;DR

This paper explores sampling and interpolation on certain non-commutative, two-step nilpotent Lie groups using time-frequency analysis, generalizing previous results from the Heisenberg group.

Contribution

It provides new sufficient conditions for sampling spaces with interpolation properties on these Lie groups, extending prior work on the Heisenberg group.

Findings

Established explicit criteria for sampling and interpolation on these groups.

Connected time-frequency analysis techniques with nilpotent Lie group theory.

Generalized results from the Heisenberg group to broader classes of Lie groups.

Abstract

Let $N$ be a non-commutative, simply connected, connected, two-step nilpotent Lie group with Lie algebra $\mathfrak{n}$ such that $\mathfrak{n=a\oplus b\oplus z}$, $\left[ \mathfrak{a},\mathfrak{b}\right] \subseteq \mathfrak{z},$ the algebras $\mathfrak{a},\mathfrak{b,z}$ are abelian, $\mathfrak{a}=\mathbb{R}\text{-span}\left\{ X_{1},X_{2},\cdots,X_{d}\right\} ,$ and $\mathfrak{b}=\mathbb{R}\text{-span}\left\{ Y_{1},Y_{2},\cdots ,Y_{d}\right\} .$ Also, we assume that $\det\left[ \left[ X_{i}% ,Y_{j}\right] \right] _{1\leq i,j\leq d}$ is a non-vanishing homogeneous polynomial in the unknowns $Z_{1},\cdots,Z_{n-2d}$ where $\left\{ Z_{1},\cdots,Z_{n-2d}\right\} $ is a basis for the center of the Lie algebra. Using well-known facts from time-frequency analysis, we provide some precise sufficient conditions for the existence of sampling spaces with the interpolation property, with respect to some discrete subset of $N$. The result obtained in this work can be seen as a direct application of time-frequency analysis to the theory of nilpotent Lie groups. Several explicit examples are computed. This work is a generalization of recent results obtained for the Heisenberg group by Currey, and Mayeli in \cite{Currey}.