### Commutation relations of the Gabor frame operator

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

 (10)

It follows that also commutes with and , so that 0

Therefore the elements of the dual Gabor frame are generated by a single function , analogously to the . This observation bears important computational advantages. To compute the dual system one computes the (canonically) dual atom and derives all other elements 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 and even different choices of 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 -norm among all possible sets of coefficients, and at the same time is the -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

 (11)

which implies that if is a frame for given parameters , then is a frame with reverse parameters .