Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle (2304.03324v1)

Abstract: Let $({f_j}{j=1}^n, {\tau_j}{j=1}^n)$ and $({g_k}{k=1}^m, {\omega_k}{k=1}^m)$ 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 |\theta_f x|0^\frac{1}{p}|\theta_g x|_0^\frac{1}{q} \geq \frac{1}{\displaystyle\max{1\leq j\leq n, 1\leq k\leq m}|f_j(\omega_k)|}\quad \text{and} \quad |\theta_g x|0^\frac{1}{p}|\theta_f x|_0^\frac{1}{q}\geq \frac{1}{\displaystyle\max{1\leq j\leq n, 1\leq k\leq m}|g_k(\tau_j)|}. \end{align} where \begin{align*} \theta_f: \mathcal{X} \ni x \mapsto (f_j(x) ){j=1}^n \in \ell^p([n]); \quad \theta_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-Torr\'{e}sani Uncertainty Principle}. Inequality (1) improves Ricaud-Torr\'{e}sani 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]}.