Functional Ghobber-Jaming Uncertainty Principle

06/01/2023

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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^n, {ω_k}_k=1^n) be two p-orthonormal bases for a finite dimensional Banach space 𝒳. Let M,N⊆{1, …, n} be such that o(M)^1/qo(N)^1/p< 1/max_1≤ j,k≤ n|g_k(τ_j) |, where q is the conjugate index of p. Then for all x ∈𝒳, we show that (1) x≤(1+1/1-o(M)^1/qo(N)^1/pmax_1≤ j,k≤ n|g_k(τ_j)|)[(∑_j∈ M^c|f_j(x)|^p)^1/p+(∑_k∈ N^c|g_k(x) |^p)^1/p]. We call Inequality (1) as Functional Ghobber-Jaming Uncertainty Principle. Inequality (1) improves the uncertainty principle obtained by Ghobber and Jaming [Linear Algebra Appl., 2011].