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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 $\ell^r(\N)$ 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.
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Submitted on : Monday, February 22, 2010 - 9:28:42 AM
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  • HAL Id : hal-00458658, version 1
  • ARXIV : 1002.4076

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

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