Parameterization of m band affine frames with pre-assigned scaling filter

Affine (wavelet) frames related with subdivision scheme provide tools for building curves and surfaces in CAGD/CAM. At first we show that scaling filters can be parameterized by decomposed elementary matrices with the form Im−P+Pz−k, where k is a nonnegative integer, and P is a one rank constant-valued idempotent matrix P2=Im (Theorem 2.2). Then we use the method to construct m band affine (wavelet) frames with the given scaling filter and the dual frames. General solutions for affine (wavelet) frames and its dual affine (wavelet) frames are obtained with pre-assigned scaling filter (Theorems 3.5 and 3.6). At last a design example is given.