Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle

04/05/2023

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by   K. Mahesh Krishna, et al.

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Let ({f_j}_j=1^n, {τ_j}_j=1^n) and ({g_k}_k=1^m, {ω_k}_k=1^m) be p-Schauder frames for a finite dimensional Banach space 𝒳. Then for every x ∈𝒳∖{0}, we show that (1) θ_f x_0^1/pθ_g x_0^1/q≥1/max_1≤ j≤ n, 1≤ k≤ m|f_j(ω_k)| and θ_g x_0^1/pθ_f x_0^1/q≥1/max_1≤ j≤ n, 1≤ k≤ m|g_k(τ_j)|. where θ_f: 𝒳∋ x ↦ (f_j(x) )_j=1^n ∈ℓ^p([n]); θ_g: 𝒳∋ x ↦ (g_k(x) )_k=1^m ∈ℓ^p([m]) and q is the conjugate index of p. We call Inequality (1) as Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle. Inequality (1) 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].