A method of construction for biorthogonal multiwavelets system with 2r multiplicity

Biorthogonal multiwavelets systems are consisted of a pair of biorthogonal multiscaling functions and the corresponding a pair of multiwavelets. A method for constructing biorthogonal multiwavelets systems with multiplicity 2r was derived. In particular Hermite interpolation mask was considered as primal masks. A pair of primal and dual masks with high order of sum rules should be firstly designed in order to obtain a biorthogonal multiwavelet with high vanishing moment. Concretely, starting from a short sequence primal masks possessing Hermite interpolation properties, a necessary and sufficient conditions such that primal masks satisfies the preassigned order of sum rules are established, then a dual masks with short support, symmetry and any preassigned order of sum rules is constructed, it follows that two multiscaling functions are obtained and then multiwavelets and dual multiwavelets are derived. From which, a general design framework is obtained for constructing biorthogonal multiwavelets system with multiplicity 2r associated with Hermite interpolation functions. The biorthogonal multiwavelets constructed by our design framework has desirable properties such as symmetry, short support, high vanishing moments and simple structures with explicit expressions. Finally, an example is given.