Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principles

Abstract

Let $(\Omega ,\mu )$, $(\Delta ,\nu )$ be measure spaces and $p=1$ or $p=\infty$. Let $(\{f_{\alpha }{\}}_{\alpha \in \Omega },\{{\tau }_{\alpha }{\}}_{\alpha \in \Omega })$ and $(\{g_{\beta }{\}}_{\beta \in \Delta },\{{\omega }_{\beta }{\}}_{\beta \in \Delta })$ be unbounded continuous p-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x\in (\mathcal{D}({\theta }_f)\cap \mathcal{D}({\theta }_g))\setminus \{0\}$, we show that \begin{align}\label{UB} (1) \quad \quad \quad \quad μ(\operatorname{supp}(θ_f x))ν(\operatorname{supp}(θ_g x)) \geq \frac{1}{\left(\displaystyle\sup_{α\in Ω, β\in Δ}|f_α(ω_β)|\right)\left(\displaystyle\sup_{α\in Ω, β\in Δ}|g_β(τ_α)|\right)}, \end{align} where \begin{align*} &θ_f:\mathcal{D}(θ_f) \ni x \mapsto θ_fx \in \mathcal{L}^p(Ω, μ); \quad θ_fx: Ω\ni α\mapsto (θ_fx) (α):= f_α(x) \in \mathbb{K},\\ &θ_g: \mathcal{D}(θ_g) \ni x \mapsto θ_gx \in \mathcal{L}^p(Δ, ν); \quad θ_gx: Δ\ni β\mapsto (θ_gx) (β):= g_β(x) \in \mathbb{K}. \end{align*} We call Inequality (1) as \textbf{Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle}. Along with recent \textbf{Functional Continuous Uncertainty Principle} [arXiv:2308.00312], Inequality (1) also improves Ricaud-Torrésani uncertainty principle [IEEE Trans. Inform. Theory, 2013]. In particular, it improves Elad-Bruckstein uncertainty principle [IEEE Trans. Inform. Theory, 2002] and Donoho-Stark uncertainty principle [SIAM J. Appl. Math., 1989].

arXiv comment6 Figures, 0 Figures