Differentiation and the Balian-Low Theorem

The Balian-Low theorem (BLT) is a key result in time-frequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system $\{e^{2\pi imbt} \, g(t-na)\}_{m,n \in \mbox{\bf Z}}$ with $ab=1$ forms an orthonormal basis for $L^2({\bf R}),$ then $\left(\int_{-\infty}^\infty |t \, g(t)|^2 \, dt\right) \, \left(\int_{-\infty}^\infty |\gamma \, \hat g(\gamma)|^2 \, d\gamma\right) = +\infty.$ The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that $(g')^{\wedge}(\gamma) = 2 \pi i \gamma \, \hat g(\gamma)$ , the role of differentiation in the proof of the BLT is examined carefully. The major new contributions of this paper are the construction of a complete Gabor system of the form $\{e^{2\pi ib_mt\} \, g(t-a_n)}$ such that $\{(a_n,b_m)\}$ has density strictly less than 1, an Amalgam BLT that provides distinct restrictions on Gabor systems $\{e^{2\pi imbt} \, g(t-na)\}$ that form exact frames, and a new proof of the BLT for exact frames that does not require differentiation and relies only on classical real variable methods from harmonic analysis.