Approximate dual Gabor atoms via the adjoint lattice method

Regular Gabor frames for L 2 ( ℝ d ) are obtained by applying time-frequency shifts from a lattice in Λ ◃ ℝ d × ℝ ̂ 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 ∈ S 0 ( ℝ d ) . There is always a canonical dual frame, generated by the dual Gabor atom g ~ . 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 a ℤ d × b ℤ d ). 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.