Functional Kuppinger-Durisi-Bölcskei Uncertainty Principle

Let $\mathcal{X}$ be a Banach space. Let $\{{\tau }_j{\}}_{j=1}^n,\{{\omega }_k{\}}_{k=1}^m\subseteq \mathcal{X}$ and $\{f_j{\}}_{j=1}^n$, $\{g_k{\}}_{k=1}^m\subseteq {\mathcal{X}}^{\ast }$ satisfy $\mathrm{\mid }{f}_j({\tau }_j)\mathrm{\mid }\ge 1$ for all $1\le j\le n$, $\mathrm{\mid }{g}_k({\omega }_k)\mathrm{\mid }\ge 1$ for all $1\le k\le m$. If $x\in \mathcal{X}\setminus \{0\}$ is such that $x={\theta }_{\tau }{\theta }_{f}x={\theta }_{\omega }{\theta }_{g}x$, then we show that \begin{align}\label{FKDB} (1) \quad\quad\quad\quad |θ_fx|0|θ_gx|0\geq \frac{\bigg[1-(|θ_fx|0-1)\max\limits{1\leq j,r \leq n,j\neq r}|f_j(τ_r)|\bigg]^+\bigg[1-(|θ_g x|0-1)\max\limits{1\leq k,s \leq m,k\neq s}|g_k(ω_s)|\bigg]^+}{\left(\displaystyle\max{1\leq j \leq n, 1\leq k \leq m}|f_j(ω_k)|\right)\left(\displaystyle\max{1\leq j \leq n, 1\leq k \leq m}|g_k(τ_j)|\right)}. \end{align} We call Inequality (1) as $\mathbf{\text{Functional}}\mathbf{\text{Kuppinger-Durisi-B}}\ddot{\mathbf{\text{o}}}\mathbf{\text{lcskei}}\mathbf{\text{Uncertainty}}\mathbf{\text{Principle}}$. Inequality (1) improves the uncertainty principle obtained by Kuppinger, Durisi and Bölcskei $\mathit{\text{[IEEE}}\mathit{\text{Trans.}}\mathit{\text{Inform.}}\mathit{\text{Theory}}\mathit{\text{(2012)]}}$ (which improved the Donoho-Stark-Elad-Bruckstein uncertainty principle $\mathit{\text{[SIAM}}\mathit{\text{J.}}\mathit{\text{Appl.}}\mathit{\text{Math.}}\mathit{\text{(1989),}}\mathit{\text{IEEE}}\mathit{\text{Trans.}}\mathit{\text{Inform.}}\mathit{\text{Theory}}\mathit{\text{(2002)]}}$). We also derive functional form of the uncertainity principle obtained by Studer, Kuppinger, Pope and Bölcskei $\mathit{\text{[EEE}}\mathit{\text{Trans.}}\mathit{\text{Inform.}}\mathit{\text{Theory}}\mathit{\text{(2012)]}}$.