Discrete transforms produced from two natural numbers and applications
EUROCAST'11 Proceedings of the 13th international conference on Computer Aided Systems Theory - Volume Part II
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We introduce a class of sparse unimodular matrices $U^m$ of order $m \times m$, $m = 2,3, \ldots\,$. 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.