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 \|θ_τx \|_0 \|θ_ωx \|_0 \geq \frac{1}{\sup_{n, m \in \mathbb{N}} \|\langle τ_n, ω_m\rangle \|^2}. \end{align} We call Inequality (1) as \textbf{Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle}. Inequality (1) is the noncommutative analogue of breakthrough Ricaud-Torrésani 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]}.

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