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

摘要

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 ${E_{Bm}T_{Cn}g}_{m,nin {Bbb Z}^d}$ 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 ${E_{Bm}T_{Cn}g}_{m,nin {Bbb Z}^d, kin {frak F}}$ , 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.