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

2023/12/01 by Krishna, K. Mahesh
#42C15 #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Information Theory (cs.IT) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2312.00366

Abstract

Let (Ω, μ), (Δ, ν) be measure spaces and p=1 or p=∞. Let (\f_α\_{α∈ Ω}, \τ_α\_{α∈ Ω}) and (\g_β\_{β∈ Δ}, \ω_β\_{β∈ Δ}) be unbounded continuous p-Schauder frames for a Banach space X. Then for every x ∈ ( D(θ_f) \capD(θ_g))∖\0\, we show that (1) μ(supp(θ_f x))ν(supp(θ_g x)) ≥ \frac1(sup_{α∈ Ω, β∈ Δ}|f_α(ω_β)|)(sup_{α∈ Ω, β∈ Δ}|g_β(τ_α)|), where amp;θ_f:D(θ_f) \ni x ↦ θ_fx ∈ L^p(Ω, μ); θ_fx: Ω\ni α↦ (θ_fx) (α):= f_α(x) ∈ \mathbbK,
amp;θ_g: D(θ_g) \ni x ↦ θ_gx ∈ L^p(Δ, ν); θ_gx: Δ\ni β↦ (θ_gx) (β):= g_β(x) ∈ \mathbbK. We call Inequality (1) as Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle. Along with recent 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].

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