Pairs of Explicitly Given Dual Gabor Frames in L²(ℝ^d)

Abstract

Given certain compactly supported functions g ≥ L²(ℝ^d) whose ℤ^d-translates form a partition of unity, and real invertible d × d matrices B,C for which ||C^T B|| is sufficiently small, we prove that the Gabor system $\{E_{Bm}T_{Cn}g\}_{m,n\in {\Bbb Z}^d}$ forms a frame, with a (noncanonical) dual Gabor frame generated by an explicitly given finite linear combination of shifts of g. For functions g of the above type and arbitrary real invertible d × d matrices B,C this result leads to a construction of a multi-Gabor frame $\{E_{Bm}T_{Cn}g\}_{m,n\in {\Bbb Z}^d, k\in {\frak F}}$, where all the generators g_k are dilated and translated versions of g. Again, the dual generators have a similar form, and are given explicitly. Our concrete examples concern box splines.