Functional Donoho-Stark Approximate Support Uncertainty Principle (2307.01215v1)

Abstract: 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 dimensional Banach 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\leq j,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 conjugate index of $p$. We call Inequalities (1) and (2) as \textbf{Functional Donoho-Stark Approximate Support Uncertainty Principle}. Inequalities (1) and (2) improve the finite approximate support uncertainty principle obtained by Donoho and Stark \textit{[SIAM J. Appl. Math., 1989]}.