2023/08/01 by Krishna, K. Mahesh
#42C15 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2308.00312
Let (Ω, μ), (Δ, ν) be measure spaces. Let (\fα\α∈ Ω, \τα\α∈ Ω) and (\gβ\β∈ Δ, \ωβ\β∈ Δ) be continuous p-Schauder frames for a Banach space X. Then for every x ∈ X∖\0\, we show that (1) μ(supp(θf x))^(1)/(p) ν(supp(θg x))^(1)/(q) ≥ \frac1supα∈ Ω, β∈ Δ|fα(ωβ)|, ν(supp(θg x))^(1)/(p) μ(supp(θf x))^(1)/(q)≥ \frac1supα∈ Ω, β∈ Δ|gβ(τα)|. where amp;θf: X \ni x ↦ θfx ∈ Lp(Ω, μ); θfx: Ω\ni α↦ (θfx) (α):= fα(x) ∈ \mathbbK, amp;θg: X \ni x ↦ θgx ∈ Lp(Δ, ν); θgx: Δ\ni β↦ (θgx) (β):= gβ(x) ∈ \mathbbK 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.