Gabor frame

Remarks

The coefficients $\ip{f}{M_{bn} T_{ak} g}$ are the values $V_g f(ak, bn)$ of the short-time Fourier transform on the lattice $a\Z \times b\Z$. If $\mathcal{G}(g, a, b)$ is a frame, then $ab \le 1$; for $ab = 1$ it is a frame only if it is a Riesz basis. For the Gaussian $g(t) = e^{-\pi t^2}$, the system $\mathcal{G}(g, a, b)$ is a frame exactly when $ab < 1$, a result of Lyubarskii and of Seip.

Example: let $g = 1_{[0, 1)}$ and $a = b = 1$. The translates $T_k g$ are the indicators of the intervals $[k, k+1)$, which tile $\R$. On each such interval the functions $e^{2\pi i n t}$, $n \in \Z$, form an orthonormal basis of $L^2([k, k+1))$. So $\mathcal{G}(g, 1, 1)$ is an orthonormal basis of $L^2(\R)$, hence a Gabor frame with $A = B = 1$.

Non-example: with $g = 1_{[0, 1)}$, $a = 2$ and $b = 1$, every function in the system vanishes on $[1, 2)$. So a nonzero $f$ supported in $[1, 2)$ has all coefficients $0$, and the lower frame bound fails.