Factoring Pseudoidentity Matrix Pairs

TL;DR

This paper revises the factorization theory of pseudoidentity matrix pairs related to biorthogonal wavelets, correcting previous conjectures and providing a new constructive method for rank 2 cases.

Contribution

It disproves a prior conjecture on rank 2 pseudoidentity matrix pairs and establishes a generalized factorization theorem for ranks m ≥ 2, including a constructive Euclidean algorithm-based method.

Findings

Disproved the conjecture on rank 2 pseudoidentity matrix pairs.

Proved a new factorization theorem valid for rank m ≥ 2.

Provided a constructive Euclidean algorithm-based method for rank 2 cases.

Abstract

The problem of factorization and parametrization of compactly supported biorthogonal wavelets was reduced to that of pseudoidentity matrix pairs by Resnikoff, Tian, and Wells in their 2001 paper. Based on a conjecture on the pseudoidentity matrix pairs of rank 2 stated in the same paper, they proved a theorem which gives a complete factorization result for rank 2 compactly supported biorthogonal wavelets. In this paper, we first provide examples to show that the conjecture is not true, then we prove a factorization theorem for pseudoidentity matrix pairs of rank $m\ge 2$. As a consequence, our result shows that a slightly modified version of the factorization theorem in the rank 2 case given by Resnikoff, Tian, and Wells holds. We also provide a concrete constructive method for the rank 2 case which is determined by applying the Euclidean algorithm to two polynomials.