Regularity for complete and minimal Gabor systems on a lattice

Abstract

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.