A class of sparse unimodular matrices generating multiresolution and sampling analysis for data of any length

We introduce a class of sparse unimodular matrices U-m of order m x m, m = 2, 3,.... Each matrix U-m has all entries 0 except for a small number of entries 1. The construction of U-m is achieved by iteration, determined by the prime factorization of a positive integer m and by new dilation operators and block matrix operators. The iteration above gives rise to a multiresolution analysis of the space V-m of all m-periodic complex-valued sequences, suitable to reveal information at different scales and providing sampling formulas on the multiresolution subspaces of V-m. We prove that the matrices U-m are invertible, and we present a recursion equation to compute the inverse matrices. Finally, we connect the transform induced by the matrix U-m with the underlying natural tree structure and random walks on trees.