2024/01/01 by K. Mahesh Krishna, Krishna, K. Mahesh
Decision Sciences · #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2402.04255
Let X be a Banach space. Let \τj\j=1n, \ωk\k=1m⊆ X and \fj\j=1n, \gk\k=1m⊆ X^* satisfy |fj(τj)|≥ 1 for all 1≤ j ≤ n, |gk(ωk)|≥ 1 for all 1≤ k ≤ m. If x ∈ X∖ \0\ is such that x=θτθf x=θωθg x, then we show that (1) ‖θfx‖0‖θgx‖0≥ \frac[1-(‖θfx‖0-1)max1≤ j,r ≤ n,j≠ r|fj(τr)|]+[1-(‖θg x‖0-1)max1≤ k,s ≤ m,k≠ s|gk(ωs)|]+(max1≤ j ≤ n, 1≤ k ≤ m|fj(ωk)|)(max1≤ j ≤ n, 1≤ k ≤ m|gk(τj)|). We call Inequality (1) as Functional Kuppinger-Durisi-Bölcskei Uncertainty Principle. Inequality (1) improves the uncertainty principle obtained by Kuppinger, Durisi and Bölcskei [IEEE Trans. Inform. Theory (2012)] (which improved the Donoho-Stark-Elad-Bruckstein uncertainty principle [SIAM J. Appl. Math. (1989), IEEE Trans. Inform. Theory (2002)]). We also derive functional form of the uncertainity principle obtained by Studer, Kuppinger, Pope and Bölcskei [EEE Trans. Inform. Theory (2012)].