Explicit construction of symmetric compactly supported biorthogonal multiwavelets via group transformations

Polyphase matrix extension of multiscaling vectors is a fundamental approach for the construction of compactly supported biorthogonal multiwavelets. In this paper, canonical form of polyphase matrices of multiscaling vectors and some properties of solutions satisfying the matrix equation of PR condition are studied through unimodular groups over the Laurent polynomial ring. Under the canonical form of polyphase matrices of multiscaling vectors with the same symmetric center and different symmetric centers, explicit symmetric extension formulas expressed by the polyphase matrices of multiscaling vectors can be obtained. As a result, a novel approach for the construction of symmetric compactly supported biorthogonal multiwavelets with multiplicity 2 is proposed. Finally, several examples are given for verification of our proposed approach.