Functional Donoho-Stark Approximate Support Uncertainty Principle

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 ${\theta }_{f}x$ is $\epsilon$-supported on $M\subseteq \{1,\dots ,n\}$ w.r.t. p-norm and ${\theta }_{g}x$ 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(ω_k) |}\max \{1-\varepsilon-δ, 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(τ_j) |}\max \{1-\varepsilon-δ, 0\},\label{ME2} \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}^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]}.

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