Convergence Rate Of Vector Subdivision Scheme And Vector Riesz Multiwavelet Bases For L₂(R)

In this paper we investigate the following three subjects:1. The convergence rate of homogeneous and nonhomogeneous multiple vector refinement equations of the formandwhere the vector of functions (?) = {(?)₁,…, (?)_r)^T belongs to (L₁(R^s))^r, a := (a(α))_{α∈Z^s}isa finitely supported sequence of r×r matrices called refinement mask and M is an s×sinteger matrix such that lim(?) M^-n = 0, g = (g₁,…,g_r)^T is a given finitely supportedfunction on R^s.2. Starting from a pair of biorthogonal refinable functions (?),(?), let m = |detM|,defineThenψandψgenerate two Riesz bases for L₂(R^s), if for anyξ∈R^s, v = 1,...,m - 1,the following conditions are satisfied:det B(ξ)≠0.3. Providing two examples to show theorem given in part 2.