Time-frequency concentration of generating systems

TL;DR

This paper establishes uncertainty principles linking time-frequency localization and coefficient stability for generating systems in , showing limitations for bases and frames, and constructing systems with bounded dispersions and means.

Contribution

It proves new uncertainty principles for generating systems and constructs examples with bounded dispersions and means, advancing understanding of time-frequency localization constraints.

Findings

Bases and frames cannot have all dispersion and mean sequences bounded.

Constructed an exact system with all four sequences bounded.

Established quantitative relations between stability and localization properties.

Abstract

Uncertainty principles for generating systems $\{e_n\}_{n=1}^{\infty} \subset \ltwo$ are proven and quantify the interplay between $\ell^r(\N)$ coefficient stability properties and time-frequency localization with respect to $|t|^p$ power weight dispersions. As a sample result, it is proven that if the unit-norm system $\{e_n\}_{n=1}^{\infty}$ is a Schauder basis or frame for $\ltwo$ then the two dispersion sequences $\Delta(e_n)$, $\Delta(\bar{e_n})$ and the one mean sequence $\mu(e_n)$ cannot all be bounded. On the other hand, it is constructively proven that there exists a unit-norm exact system $\{f_n\}_{n=1}^{\infty}$ in $\ltwo$ for which all four of the sequences $\Delta(f_n)$, $\Delta(\bar{f_n})$, $\mu(f_n)$, $\mu(\bar{f_n})$ are bounded.