Illinois Journal of Mathematics

Regularity for complete and minimal Gabor systems on a lattice

Christopher Heil and Alexander M. Powell

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Nonsymmetrically weighted extensions of the Balian--Low theorem are proved for Gabor systems $\mathcal{G}(g,1,1)$ that are complete and minimal in ${L^2(\mathbb{R})}$. For $g\in{L^2(\mathbb{R})}$, it is proved that if $3 \lt p \leq4 \leq q \lt \infty$ satisfy $3/p + 1/q = 1$ and $\int|x|^p |g(x)|^2 \, dx \lt \infty$ and $\int|\xi|^q |\widehat{g}(\xi)|^2 \, d\xi \lt \infty$ then $\mathcal{G}(g,1,1) = \{e^{2\pi i n x} g(x-k)\}_{k,n \in{\mathbb{Z}}}$ cannot be complete and minimal in ${L^2(\mathbb{R})}$. For the endpoint case $(p,q)=(3,\infty)$, it is proved that if $g\in{L^2(\mathbb{R})}$ is compactly supported and $\int|\xi|^3 |\widehat{g}(\xi)|^2 \, d\xi \lt \infty$ then $\mathcal{G}(g,1,1)$ is not complete and minimal in ${L^2(\mathbb{R})}$. These theorems extend the work of Daubechies and Janssen from the case $(p,q)=(4,4)$. Further refinements and optimal examples are also provided.

Article information

Illinois J. Math., Volume 53, Number 4 (2009), 1077-1094.

First available in Project Euclid: 22 November 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42C15: General harmonic expansions, frames 42C25: Uniqueness and localization for orthogonal series
Secondary: 46C15: Characterizations of Hilbert spaces


Heil, Christopher; Powell, Alexander M. Regularity for complete and minimal Gabor systems on a lattice. Illinois J. Math. 53 (2009), no. 4, 1077--1094. doi:10.1215/ijm/1290435340.

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