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

K. Mahesh Krishna

doi:10.20944/preprints202406.0173.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 1 June 2024 / Approved: 4 June 2024 / Online: 5 June 2024 (02:47:21 CEST)

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}\label{UE} \|\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 (\ref{UE}) as \textbf{Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle}. Inequality (\ref{UE}) is the noncommutative analogue of breakthrough Ricaud-Torr\'{e}sani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, Inequality (\ref{UE}) extends Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}.

uncertainty principle; parseval frame; Hilbert C*-module

Computer Science and Mathematics, Analysis

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

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