Polyphase Matrix Extension for Biorthogonal Multiwavelets with Closed-Form Expressions

Polyphase matrix extension of scaling vectors is a fundamental approach for the construction of compactly supported biorthogonal multiwavelets, but at the expense of high computations. In this paper, a novel 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 vectors 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.