Summary: We introduce a class of sparse unimodular matrices \(U^m\) of order \(m \times m, m = 2,3, \cdots\,\). 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.