Explicit construction of compactly supported biorthogonal multiwavelets based on the matrix extension

Polyphase matrix extension of the scaling vector functions plays an important role in the construction of compactly supported biorthogonal multiwavelets. However, the involved computations are rather complex, and there is no unified, direct formula available so far. In this paper, by studying the canonical forms and the product preserving transformations of the polyphase matrices, an abstract algebraic approach for the matrix extension problem is proposed. More important, explicit formulas for the construction problem are represented via the submatrices of the polyphase matrices of scaling vector functions directly. Furthermore, complete solution set can be obtained from these explicit formulas via product preserving transformations. Computational examples demonstrate that by using the explicit formulas, our matrix extension algorithm is direct and effective.