Time-frequency concentration of generating systems

Abstract

Uncertainty principles for generating systems $\{e_n\}_{n=1}^{\infty} \subset \ltwo$ are proven and quantify the interplay between ${\ell }^r(\mathbb{N})$ 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 $\Delta (e_n)$, $\Delta (\overline{e_n})$ 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 $\Delta (f_n)$, $\Delta (\overline{f_n})$, $\mu (f_n)$, $\mu (\overline{f_n})$ are bounded.

14 pages