Approximate dual Gabor atoms via the adjoint lattice method

Abstract

Regular Gabor frames for ${\boldsymbol {L}{^{2}}(\mathbb {R}^d)}$ are obtained by applying time-frequency shifts from a lattice in $\boldsymbol {\Lambda } \vartriangleleft {\mathbb {R}^{d} \times \mathbb {\widehat {R}}}$ 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 $g \in {\boldsymbol {S}_{0}(\mathbb {R}^{d})}$. There is always a canonical dual frame, generated by the dual Gabor atom ${\widetilde g}$. 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 ${a\mathbb {Z}^{d}\,{\times }\,b\mathbb {Z}^{d}}$). 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 $\boldsymbol {\Lambda }\circ$. The effectiveness of this approach is justified by a new theoretical argument and demonstrated by numerical examples.