Functional Kuppinger-Durisi-Bölcskei Uncertainty Principle

Abstract

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 $|f_j({\tau }_j)|\ge 1$ for all $1\le j\le n$, $|g_k({\omega }_k)|\ge 1$ 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 \textbf{Functional Kuppinger-Durisi-Bölcskei Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Kuppinger, Durisi and Bölcskei \textit{[IEEE Trans. Inform. Theory (2012)]} (which improved the Donoho-Stark-Elad-Bruckstein uncertainty principle \textit{[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 \textit{[EEE Trans. Inform. Theory (2012)]}.

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