2023/07/01 by K. Mahesh Krishna, Krishna, K. Mahesh
Decision Sciences · Engineering · Mathematics · #42C15 #46B03 #46B04 #Advanced Numerical Methods in Computational Mathematics #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Information Theory (cs.IT) #Numerical methods in inverse problems #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2307.01215
openalex publication_date 2023/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (\fj\j=1n, \τj\j=1n) and (\gk\k=1n, \ωk\k=1n) be two p-orthonormal bases for a finite dimensional Banach space X. If x ∈ X∖\0\ is such that θfx is ε-supported on M⊆ \1,…, n\ w.r.t. p-norm and θgx is δ-supported on N⊆ \1,…, n\ w.r.t. p-norm, then we show that (1) amp;o(M)^(1)/(p)o(N)^(1)/(q)≥ \frac1 max1≤ j,k≤ n|fj(ωk) |max \1-ε-δ, 0\,
(2) \quadamp;o(M)^(1)/(q)o(N)^(1)/(p)≥ \frac1 max1≤ j,k≤ n|gk(τj) |max \1-ε-δ, 0\, where θf: X \ni x ↦ (fj(x) )j=1n ∈ ℓp([n]); θg: X \ni x ↦ (gk(x) )k=1n ∈ ℓp([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].