Approximate dual Gabor atoms via the adjoint lattice method

Regular Gabor frames for L2(Rd) are obtained by applying time-frequency shifts from a lattice in Λ Rd× R̂d to some decent so-called Gabor atom g, which typically is something like a summability kernel in classical analysis, or a Schwartz function, or more generally some g∈ S0(R). There is always a canonical dual frame, generated by the dual Gabor atom g̃. The paper promotes a numerical approach for the efficient calculation of good approximations to the dual Gabor atom for general lattices, including the non-separable ones (different from aZd×bZd). The theoretical foundation for the approach is the well-known Wexler-Raz biorthogonality relation and the more recent theory of localized frames. The combination of these principles guarantees that the dual Gabor atom can be approximated by a linear combination of a few time-frequency shifted atoms from the adjoint lattice Λ ◦. The effectiveness of this approach is justified by a new theoretical argument and demonstrated by numerical examples.