Non-Haar MRA on local fields of positive characteristic

TL;DR

This paper introduces a novel method for constructing non-Haar multiresolution analyses on local fields of positive characteristic using vector space structures and tree-based scaling functions.

Contribution

It presents a new approach to build orthogonal MRA on local fields by utilizing vector space representations and tree structures for scaling functions.

Findings

Constructed integral periodic masks for non-Haar MRA

Developed a method to generate scaling functions via trees with p^s vertices

Identified a large family of such trees for given prime p

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.