Armlets and balanced multiwavelets

Abstract

In the scalar-valued setting, it is well-known that the two-scale sequences { q k } of Daubechies orthogonal wavelets can be given explicitly by the two-scale sequences { p k } of their corresponding orthogonal scaling functions, such as q k = (-1) k p 1- k . However, due to the non-commutativity of matrix multiplication, there is little such development in the multi-wavelet literature to express the two-scale matrix sequence { Q k } of an orthogonal multi-wavelet in terms of the two-scale matrix sequence { P k } of its corresponding multi-scaling function vector. This paper, in part, is devoted to this study for the setting of orthogonal multi-wavelets of dimension r = 2. We will apply our results to constructing a family of the most recently introduced notion of armlet of order n and a family of the n -balanced orthogonal multi-wavelets.