Construction of Compactly Supported Biorthogonal Multiwavelets: An Abstract Algebraic Approach

Abstract

The polyphase matrix extension of the scaling vector functions is a fundamental approach to construction of compactly supported biorthogonal multiwavelets, but at the expense of high computations. In this paper, a novel abstract algebraic approach through factorization of the unimodular matrix over the Laurent polynomial ring is developed so that closed-form solution can be obtained for the construction of compactly supported biorthogonal multiwavelets from the scaling vector functions based on some conditions. Moreover, the relationship between any two different extensions for the same scaling vector functions can be obtained from one to another via finite steps of the product-preserving transformations, which leads to a complete solution set for the polyphase matrix extension problem.