Functional Kuppinger-Durisi-Bölcskei Uncertainty Principle (2402.04255v1)

Abstract: Let $\mathcal{X}$ be a Banach space. Let ${\tau_j}{j=1}^n, {\omega_k}{k=1}^m\subseteq \mathcal{X}$ and ${f_j}{j=1}^n$, ${g_k}{k=1}^m\subseteq \mathcal{X}^*$ satisfy $ |f_j(\tau_j)|\geq 1$ for all $ 1\leq j \leq n$, $|g_k(\omega_k)|\geq 1 $ for all $1\leq k \leq m$. 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 |\theta_fx|0|\theta_gx|_0\geq \frac{\bigg[1-(|\theta_fx|_0-1)\max\limits{1\leq j,r \leq n,j\neq r}|f_j(\tau_r)|\bigg]^+\bigg[1-(|\theta_g x|0-1)\max\limits{1\leq k,s \leq m,k\neq s}|g_k(\omega_s)|\bigg]^+}{\left(\displaystyle\max_{1\leq j \leq n, 1\leq k \leq m}|f_j(\omega_k)|\right)\left(\displaystyle\max_{1\leq j \leq n, 1\leq k \leq m}|g_k(\tau_j)|\right)}. \end{align} We call Inequality (1) as \textbf{Functional Kuppinger-Durisi-B\"{o}lcskei Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Kuppinger, Durisi and B\"{o}lcskei \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 B\"{o}lcskei \textit{[EEE Trans. Inform. Theory (2012)]}.