On The Global Support Of Scaling Vectors

A vector-valued functionφ(x) = (φ₁(x),φ₂(x),…,φ_r(x))^T(x∈R), is said to be a scaling vector if it is compactly supported and satisfies following matrix refinementequationφ(x) = sum from n=0 to N C_nφ(2x-n) with N≥1.D.K.Ruch, W.So and J.Wang [Appl. Comp. Harmonic Anal. 5(1998), 493-498] have addressed the relationship between convex support and globally linear independence of ascaling vector and find a sufficient and necessary conditions for supp (φ) = [0, N] if andonly if C₀ and C_N are not nilpotent, whenφ(x) is a globally linearly independentscaling vector.In this paper we will extend their result to the case that C₀ or C_N is nilpotent andfind a sufficient and necessary conditions for supp(φ) =[1/2^m-1,N].The obtainedresults can be viewed as supplements to the work of D. K. Ruch, W. So and J. Wang.