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Let $(\{f_j\}_{j=1}^n, \{\tau_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^n,\{\omega_k\}_{k=1}^n)$ be two p-orthonormal bases for a finite dimensionalBanach space $\mathcal{X}$. If $ x \in \mathcal{X}\setminus\{0\}$ is such that$\theta_fx$ is $\varepsilon$-supported on $M\subseteq \{1,\dots, n\}$ w.r.t.p-norm and $\theta_gx$ is $\delta$-supported on $N\subseteq \{1,\dots, n\}$w.r.t. p-norm, then we show that \begin{align}\label{ME} (1) \quad \quad \quad\quad &o(M)^\frac{1}{p}o(N)^\frac{1}{q}\geq \frac{1}{\displaystyle \max_{1\leqj,k\leq n}|f_j(\omega_k) |}\max \{1-\varepsilon-\delta, 0\},\\ (2) \quad \quad\quad \quad&o(M)^\frac{1}{q}o(N)^\frac{1}{p}\geq \frac{1}{\displaystyle\max_{1\leq j,k\leq n}|g_k(\tau_j) |}\max \{1-\varepsilon-\delta,0\},\label{ME2} \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}^n \in \ell^p([n]) \end{align*} and $q$ is the conjugateindex of $p$. We call Inequalities (1) and (2) as \textbf{FunctionalDonoho-Stark Approximate Support Uncertainty Principle}. Inequalities (1) and(2) improve the finite approximate support uncertainty principle obtained byDonoho and Stark \textit{[SIAM J. Appl. Math., 1989]}.

Functional Donoho-Stark Approximate Support Uncertainty Principle