Multiscale Haar Orthonormal Matrices with the Corresponding Riesz Products and a Characterization of Cantor-Type Languages

Abstract

We introduce a class of multiscale orthonormal matrices H(m) of order m×m, m = 2, 3,... . For m = 2 N, N = 1, 2,..., we get the well known Haar wavelet system. The term "multiscale" indicates that the construction of H(m) is achieved in different scales by an iteration process, determined through the prime integer factorization of m and by repetitive dilation and translation operations on matrices. The new Haar transforms allow us to detect the underlying ergodic structures on a class of Cantor-type sets or languages. We give a sufficient condition on finite data of lengthm, or step functions determined on the intervals [k/m, (k + 1)/m) , k = 0,...,m − 1 of [0, 1), to be written as a Riesz-type product in terms of the rows of H(m). This allows us to approximate in the weak-* topology continuous measures by Riesz-type products.