Operator-Valued Frames for the Heisenberg Group

TL;DR

This paper extends classical frame results from real line characters to operator-valued frames on the Heisenberg group, using the Selberg Trace Formula to analyze perturbations and construct decompositions in $L^2$ spaces.

Contribution

It introduces operator-valued frames for the Heisenberg group and demonstrates their stability under perturbations using advanced harmonic analysis techniques.

Findings

Perturbations of operator-valued frames remain valid under certain conditions.

The Selberg Trace Formula is effective in analyzing frame stability on the Heisenberg group.

Decomposition of $L^2$ functions via group representations is achievable with these frames.

Abstract

A classical result of Duffin and Schaeffer gives conditions under which a discrete collection of characters on $\mathbb{R}$, restricted to $E = (-1/2, 1/2)$, forms a Hilbert-space frame for $L^2(E)$. For the case of characters with period one, this is just the Poisson Summation Formula. Duffin and Schaeffer show that perturbations preserve the frame condition in this case. This paper gives analogous results for the real Heisenberg group $H_n$, where frames are replaced by operator-valued frames. The Selberg Trace Formula is used to show that perturbations of the orthogonal case continue to behave as operator-valued frames. This technique enables the construction of decompositions of elements of $L^2(E)$ for suitable subsets $E$ of $H_n$ in terms of representations of $H_n$.