Reproducing formulas associated to translation generated systems on Nilpotent Lie Groups

TL;DR

This paper characterizes continuous reproducing formulas for translation-generated systems on nilpotent Lie groups, extending discrete frame results to integral representations and applying findings to the Heisenberg group and Gabor systems.

Contribution

It provides the first characterization of continuous reproducing formulas for translation systems on nilpotent Lie groups, broadening the understanding from discrete to integral frameworks.

Findings

Derived integral reproducing formulas for nilpotent Lie groups

Extended results to the Heisenberg group and Gabor systems

Established conditions for continuous frames in $L^2(G)$

Abstract

Let $G$ be a connected, simply connected, nilpotent Lie group whose irreducible unitary representations are square-integrable modulo the center. We obtain characterization results for reproducing formulas associated with the left translation generated systems in $ L^2(G)$. Unlike the previous study of discrete frames on the nilpotent Lie groups, the current research occurs within the set up of continuous frames, which means the resulting reproducing formulas are given in terms of integral representations instead of discrete sums. As a consequence of our results for the Heisenberg group, a reproducing formula associated with the orthonormal Gabor systems of $L^2(\mathbb R^d)$ is obtained.