Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866929453609779200 |
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| author | Krishna, K. Mahesh |
| author_facet | Krishna, K. Mahesh |
| contents | Let $\{τ_n\}_{n=1}^\infty$ and $\{ω_m\}_{m=1}^\infty$ 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]}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08504 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle Krishna, K. Mahesh Operator Algebras Functional Analysis 42C15, 46L08 Let $\{τ_n\}_{n=1}^\infty$ and $\{ω_m\}_{m=1}^\infty$ 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]}. |
| title | Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle |
| topic | Operator Algebras Functional Analysis 42C15, 46L08 |
| url | https://arxiv.org/abs/2406.08504 |