Non-Haar MRA on local fields of positive characteristic

Abstract

We propose a simple method to construct integral periodic mask and corresponding scaling step functions that generate non-Haar orthogonal MRA on the local field F (s) of positive characteristic p. To construct this mask we use two new ideas. First, we consider local field as vector space over the finite field GF (p s ). Second, we construct scaling function by arbitrary tree that has p s vertices. By fixed prime number p there exist p s(p s −2) such trees.

On orthogonal systems of shifts of scaling function on local fields of positive characteristic

“…Thus, substituting (13) and using the induction hypothesis we obtain a1,0,...,0 = λ a k ,...,a1,0,...,0 λ a k−1 ,...,a1,0,...,0 . .…”

“…As was already mentioned in the introduction, a method for construction of a scaling function that generates non-Haar orthogonal MRA was specified in [13]. It is constructed by the means of some tree and results in a function such that |φ| takes two values only: 0 and 1.…”

“…Now we prove that if any vertex of level l = k − 1 < N satisfies the condition a (0) a k−2 ,...,a1,0,...,0 = 1 , then such a condition is also satisfied by any vertex of level l = k ≤ N of the treeT . Using (13) and substituting…”

“…In these articles only Haar wavelets were obtained. In the article by Lukomskii and Vodolazov [13], another method to construct integral periodic masks and corresponding scaling step functions that generate non-Haar orthogonal MRA were developed.…”

“…However, in the article [13], only the simple case of mask m 0 being elementary was considered, i.e. m 0 (χ)…”