Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle

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 |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 $\mathbf{\text{Functional}}\mathbf{\text{Donoho-Stark-Elad-Bruckstein-Ricaud-Torr}}\acute{\mathbf{\text{e}}}\mathbf{\text{sani}}\mathbf{\text{Uncertainty}}\mathbf{\text{Principle}}$. Inequality (1) improves Ricaud-Torrésani uncertainty principle $\mathit{\text{[IEEE}}\mathit{\text{Trans.}}\mathit{\text{Inform.}}\mathit{\text{Theory,}}\mathit{\text{2013]}}$. In particular, it improves Elad-Bruckstein uncertainty principle $\mathit{\text{[IEEE}}\mathit{\text{Trans.}}\mathit{\text{Inform.}}\mathit{\text{Theory,}}\mathit{\text{2002]}}$ and Donoho-Stark uncertainty principle $\mathit{\text{[SIAM}}\mathit{\text{J.}}\mathit{\text{Appl.}}\mathit{\text{Math.,}}\mathit{\text{1989]}}$.