Functional Continuous Uncertainty Principle

08/01/2023

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by   K. Mahesh Krishna, et al.

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Let (Ω, μ), (Δ, ν) be measure spaces. Let ({f_α}_α∈Ω, {τ_α}_α∈Ω) and ({g_β}_β∈Δ, {ω_β}_β∈Δ) be continuous p-Schauder frames for a Banach space 𝒳. Then for every x ∈𝒳∖{0}, we show that (1) μ(supp(θ_f x))^1/pν(supp(θ_g x))^1/q≥1/sup_α∈Ω, β∈Δ|f_α(ω_β)|, ν(supp(θ_g x))^1/pμ(supp(θ_f x))^1/q≥1/sup_α∈Ω , β∈Δ|g_β(τ_α)|. where θ_f: 𝒳∋ x ↦θ_fx ∈ℒ^p(Ω, μ); θ_fx: Ω∋α↦ (θ_fx) (α):= f_α (x) ∈𝕂, θ_g: 𝒳∋ x ↦θ_gx ∈ℒ^p(Δ, ν); θ_gx: Δ∋β↦ (θ_gx) (β):= g_β (x) ∈𝕂 and q is the conjugate index of p. We call Inequality (1) as 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.