The Balian-Low theorem regularity of Gabor systems
The Gabor system generated by a function \(g\in L^2(\mathbb{R})\) is the collection of functions defined by \( \{e^{2\pi imx}g(x-n)\}_{m,n\in \mathbb{Z}}\). The well-known Balian-Low theorem states that if this Gabor system forms an orthonormal basis for \(L^2(\mathbb{R})\), then \( \int | xg(x)| ^2dx = \infty\) or \(\int | \gamma \hat{g}(\gamma)| ^2d\gamma =\infty\). Thus, a function \(g\) generating a Gabor orthonormal basis can not be well localized in both time and frequency. The main result of the paper proves, for any \(\varepsilon>0\) and \(A,B>0\) with \(1/A+1/B=1\), the existence of a Gabor orthonormal basis for which \( \int | g(x)| ^2\frac{1+ | x| ^A}{\log^{1+\varepsilon}(2+| x| )}dx <\infty\) and \(\int | \hat{g}(\gamma)| ^2\frac{1+ | \gamma| ^B}{\log^{2+\varepsilon}(2+| \gamma | )}d\gamma <\infty.\)
scientific article; zbMATH DE number 1983429
scientific article