Time-frequency concentration of generating systems

Philippe Jaming (1, 2) , Alexander M. Powell (3)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 IMB - Institut de Mathématiques de Bordeaux
3 Department of Mathematics, Vanderbilt University

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 $\Delta(e_n)$, $\Delta(\widehat{e_n})$ 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 $\Delta(f_n)$, $\Delta(\widehat{f_n})$, $\mu(f_n)$, $\mu(\widehat{f_n})$ are bounded.

Mots clés: Compactness; exact system; frame; Schauder basis; time-frequency concentration; uncertainty principle

Domaines: Analyse classique [math.CA]

Philippe Jaming, Alexander M. Powell. Time-frequency concentration of generating systems. Proceedings of the American Mathematical Society, 2011, 139, pp.3279-3290. ⟨hal-00458658⟩