Fast transforms: Banded matrices with banded inverses

It is unusual for both A and A-1 to be banded—but this can be a valuable property in applications. Block-diagonal matrices F are the simplest examples; wavelet transforms are more subtle. We show that every example can be factored into A = F1…FN where N is controlled by the bandwidths of A and A-1 (but not by their size, so this extends to infinite matrices and leads to new matrix groups).