Functional Kuppinger-Durisi-Bölcskei Uncertainty Principle
K. Mahesh Krishna
TL;DR
The paper develops a functional generalization of uncertainty principles within Banach spaces by introducing the operator families $\theta_f$, $\theta_\tau$, $\theta_g$, and $\theta_\omega$ built from functionals and elements. It proves a finite-dimensional Functional Kuppinger-Durisi-Bölcskei Uncertainty Principle (FKDB) that bounds the product of sparsities $\|\theta_f x\|_0$ and $\|\theta_g x\|_0$ with a bound depending on cross-correlations $|f_j(\tau_r)|$ and $|g_k(\omega_s)|$, improving prior results and linking to Studer–Kuppinger–Pope–Bölcskei. It then extends these ideas to infinite dimensions via 1-approximate Bessel sequences (1-ABS), establishing infinite-dimensional FKDB and SKPB-type bounds using suprema rather than maxima and clarifying conditions for boundedness of the associated synthesis and analysis maps. The work also discusses potential continuous versions (Lebesgue spaces) and poses open questions regarding dimension-divisor refinements and extensions to other algebraic settings, highlighting the broad applicability of functional uncertainty principles in analysis and signal processing.
Abstract
Let $\mathcal{X}$ be a Banach space. Let $\{τ_j\}_{j=1}^n, \{ω_k\}_{k=1}^m\subseteq \mathcal{X}$ and $\{f_j\}_{j=1}^n$, $\{g_k\}_{k=1}^m\subseteq \mathcal{X}^*$ satisfy $ |f_j(τ_j)|\geq 1$ for all $ 1\leq j \leq n$, $|g_k(ω_k)|\geq 1 $ for all $1\leq k \leq m$. If $x \in \mathcal{X}\setminus \{0\}$ is such that $x=θ_τθ_f x=θ_ωθ_g x$, 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)]}.