Abstract
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 \(\{E_{Bm}T_{Cn}g\}_{m,n\in {\Bbb Z}^d}\) 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 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.
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Christensen, O., Kim, R. Pairs of Explicitly Given Dual Gabor Frames in L2(ℝd). J Fourier Anal Appl 12, 243–255 (2006). https://doi.org/10.1007/s00041-005-5052-3
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DOI: https://doi.org/10.1007/s00041-005-5052-3