Pairs of explicitly given dual Gabor frames in L 2 ( R d )

Given certain compactly supported functions g ∈ L2(Rd) whose Zd-translates form a partition of unity, and real invertible d×d matrices B, C for which ||CT B|| is sufficiently small, we prove that the Gabor system {EBmTCng}m,n∈Zd forms a frame, with a (non-canonical) 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 {EBmTCngk}m,n∈Zd,k∈F , where all the generators gk 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.