Commutation relations of the Gabor frame operator

One can easily check (see also [Dau90]) that the Gabor frame operator commutes with translations by $ a$ and modulations by $ b$, i.e.,

$\displaystyle S T_a = T_a S \,, \quad S M_b = M_b S \,.$ (10)

It follows that $ S^{-1}$ also commutes with $ T_a$ and $ M_b$, so that 0

$\displaystyle \gamma_{m,n}$ $\displaystyle = S^{-1} g_{m,n}= S^{-1} T_{na} M_{mb} g$    
  $\displaystyle = T_{na} M_{mb} S^{-1} g = T_{na} M_{mb} \gamma \,.$    

$\displaystyle \gamma_{m,n}= S^{-1} g_{m,n}= S^{-1} T_{na} M_{mb} g= T_{na} M_{mb} S^{-1} g = T_{na} M_{mb} \gamma \,.$    

Therefore the elements of the dual Gabor frame $ \{\gamma_{m,n}\}$ are generated by a single function $ \gamma$, analogously to the $ g_{m,n}$. This observation bears important computational advantages. To compute the dual system $ \{\gamma_{m,n}\}$ one computes the (canonically) dual atom $ \gamma = S^{-1} g$ and derives all other elements $ \gamma_{m,n}$ of the dual frame by translations and modulations.

Clearly, since the elements of a frame are in general linear dependent, there are many choices for the coefficients $ c_{m,n}$ and even different choices of $ \gamma$ are possible. However, speaking with the words of I. Daubechies, the coefficients determined by the dual frame, are the most economical ones, in the sense that they have minimal $ \ltsp$-norm among all possible sets of coefficients, and at the same time $ \gamma$ is the $ \Ltsp$-function with minimal norm for which (0.8) is valid.

0 Due to the duality of translation and modulation, Gabor frames exhibit a symmetry under Fourier transform

$\displaystyle \hat{(g_{m,n})}(\omega) = \hat{g}(\omega - mb) e^{-2\pi i \omega na}$ (11)

which implies that if $ \{g_{m,n}\}$ is a frame for given parameters $ a,b$, then $ \{\hat{g_{m,n}}\}$ is a frame with reverse parameters $ b,a$.