Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle

TL;DR

This paper establishes a new noncommutative uncertainty principle for Hilbert C*-modules, extending classical results by Elad-Bruckstein, Donoho-Stark, and Ricaud-Torrésani to a broader algebraic setting.

Contribution

It introduces a noncommutative analogue of classical uncertainty principles using modular Parseval frames in Hilbert C*-modules, generalizing previous results.

Findings

Established a lower bound involving frame inner products.

Extended classical uncertainty principles to noncommutative settings.

Provided a new inequality applicable to Hilbert C*-modules.

Abstract

Let $\{\tau_n\}_{n=1}^\infty$ and $\{\omega_m\}_{m=1}^\infty$ be two modular Parseval frames for a Hilbert C*-module $\mathcal{E}$. Then for every $x \in \mathcal{E}\setminus\{0\}$, we show that \begin{align} (1) \quad \quad \quad \quad \|\theta_\tau x \|_0 \|\theta_\omega x \|_0 \geq \frac{1}{\sup_{n, m \in \mathbb{N}} \|\langle \tau_n, \omega_m\rangle \|^2}. \end{align} We call Inequality (1) as \textbf{Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle}. Inequality (1) is the noncommutative analogue of breakthrough Ricaud-Torr\'{e}sani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, Inequality (1) extends Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}.