Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918415366619136 |
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| author | Krishna, K. Mahesh |
| author_facet | Krishna, K. Mahesh |
| contents | Let $(\{f_j\}_{j=1}^n, \{τ_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^m, \{ω_k\}_{k=1}^m)$ be p-Schauder frames for a finite dimensional Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align} (1) \quad \|θ_f x\|_0^\frac{1}{p}\|θ_g x\|_0^\frac{1}{q} \geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|f_j(ω_k)|}\quad \text{and} \quad \|θ_g x\|_0^\frac{1}{p}\|θ_f x\|_0^\frac{1}{q}\geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|g_k(τ_j)|}. \end{align} where \begin{align*} θ_f: \mathcal{X} \ni x \mapsto (f_j(x) )_{j=1}^n \in \ell^p([n]); \quad θ_g: \mathcal{X} \ni x \mapsto (g_k(x) )_{k=1}^m \in \ell^p([m]) \end{align*} and $q$ is the conjugate index of $p$. We call Inequality (1) as \textbf{Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle}. Inequality (1) improves Ricaud-Torrésani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, it improves Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_03324 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle Krishna, K. Mahesh Functional Analysis Information Theory 42C15 Let $(\{f_j\}_{j=1}^n, \{τ_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^m, \{ω_k\}_{k=1}^m)$ be p-Schauder frames for a finite dimensional Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align} (1) \quad \|θ_f x\|_0^\frac{1}{p}\|θ_g x\|_0^\frac{1}{q} \geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|f_j(ω_k)|}\quad \text{and} \quad \|θ_g x\|_0^\frac{1}{p}\|θ_f x\|_0^\frac{1}{q}\geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|g_k(τ_j)|}. \end{align} where \begin{align*} θ_f: \mathcal{X} \ni x \mapsto (f_j(x) )_{j=1}^n \in \ell^p([n]); \quad θ_g: \mathcal{X} \ni x \mapsto (g_k(x) )_{k=1}^m \in \ell^p([m]) \end{align*} and $q$ is the conjugate index of $p$. We call Inequality (1) as \textbf{Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle}. Inequality (1) improves Ricaud-Torrésani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, it improves Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}. |
| title | Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle |
| topic | Functional Analysis Information Theory 42C15 |
| url | https://arxiv.org/abs/2304.03324 |