Functional Donoho-Stark Approximate Support Uncertainty Principle
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Let ({f_j}_j=1^n, {τ_j}_j=1^n) and ({g_k}_k=1^n, {ω_k}_k=1^n) be two p-orthonormal bases for a finite dimensional Banach space 𝒳. If x ∈𝒳∖{0} is such that θ_fx is ε-supported on M⊆{1,…, n} w.r.t. p-norm and θ_gx is δ-supported on N⊆{1,…, n} w.r.t. p-norm, then we show that (1) o(M)^1/po(N)^1/q≥1/max_1≤ j,k≤ n|f_j(ω_k) |max{1-ε-δ, 0}, (2) o(M)^1/qo(N)^1/p≥1/max_1≤ j,k≤ n|g_k(τ_j) |max{1-ε-δ, 0}, where θ_f: 𝒳∋ x ↦ (f_j(x) )_j=1^n ∈ℓ^p([n]); θ_g: 𝒳∋ x ↦ (g_k(x) )_k=1^n ∈ℓ^p([n]) and q is the conjugate index of p. We call Inequalities (1) and (2) as Functional Donoho-Stark Approximate Support Uncertainty Principle. Inequalities (1) and (2) improve the finite approximate support uncertainty principle obtained by Donoho and Stark [SIAM J. Appl. Math., 1989].
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