On a family of biorthogonal armlets with prescribed properties

Abstract

The notion of armlets (Analysis-Ready multiwavelets) was introduced and studied by Jian-ao Lian most recently as a precise formulation of orthonormal multiwavelets that guarantee wavelet decomposition with highpass output not being effected by polynomial perturbation of the input. In this paper, we first extend this concept to the biorthogonal setting, and then we concern ourselves about a particular construction of biorthogonal multiscaling function and multiwavelets with multiplicity r = 2. In particular, we consider the two pairs of lowpass filters are flipped and the two pairs of highpass filters are linearphase respectively and we also give several equivalent forms of flipping property, in terms of symbols and filters associated with multiscaling functions and multiwavelets. By requiring flipping property, we can represent all the entries of the matrix symbols P ( z ) , Q ( z ) , P ˜ ( z ) , Q ˜ ( z ) of a biorthogonal multiwavelet using P 11( z) and P ˜ 11 ( z ) which are the first row and first column entries of P( z) and P ˜ ( z ) , respectively. These results will lead to a procedure for the construction of biorthogonal armlets with prescribed properties. Consequently, a family of biorthogonal multiscaling functions which are mirrored copies each other and biorthogonal multiwavelets which are armlets of higher order with symmetry and antisymmetry are constructed explicitly.