Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle

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]}.