Functional Continuous Uncertainty Principle

Abstract

Let $(\Omega ,\mu )$, $(\Delta ,\nu )$ be measure spaces. Let $(\{f_{\alpha }{\}}_{\alpha \in \Omega },\{{\tau }_{\alpha }{\}}_{\alpha \in \Omega })$ and $(\{g_{\beta }{\}}_{\beta \in \Delta },\{{\omega }_{\beta }{\}}_{\beta \in \Delta })$ be continuous p-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x\in \mathcal{X}\setminus \{0\}$, we show that \begin{align} (1) \quad \quad \quad \quad μ(\operatorname{supp}(θ_f x))^\frac{1}{p} ν(\operatorname{supp}(θ_g x))^\frac{1}{q} \geq \frac{1}{\displaystyle\sup_{α\in Ω, β\in Δ}|f_α(ω_β)|}, \quad ν(\operatorname{supp}(θ_g x))^\frac{1}{p} μ(\operatorname{supp}(θ_f x))^\frac{1}{q}\geq \frac{1}{\displaystyle\sup_{α\in Ω, β\in Δ}|g_β(τ_α)|}. \end{align} where \begin{align*} &θ_f: \mathcal{X} \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{X} \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*} and $q$ is the conjugate index of $p$. We call Inequality (1) as \textbf{Functional Continuous Uncertainty Principle}. It improves the Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle obtained by K. Mahesh Krishna in [arXiv:2304.03324v1 [math.FA], 5 April 2023]. It also answers a question asked by Prof. Philip B. Stark to the author. Based on Donoho-Elad Sparsity Theorem, we formulate Measure Minimization Conjecture.

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