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 length m, 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.

On a large class of non-linear coding methods based on Boolean invertible matrices

“…In [3] and [4], we introduced the discrete RP transform with respect to a class of orthonormal matrices H(m) of order m × m. The particular matrices H(m) may be considered as a generalization of the usual Haar matrices, since their construction was based on dilation and translation operations on matrices and every row of H(m) is an unbalanced Haar function.…”