Characterization Of The Existence Of Solutions Of Multivariate Vector Refinement Equations

In this paper we investigate the multivariate vector refinement equations is the formwhere the vector of functionsφ= (φ₁,…,φ_m)^T is in (L_p(R^d))^m, (1≤p≤∞), a = (a(α))_{α∈Z^d} is m×m matrices and Mis an d×d integer matrix such that lim_{n→∞}M^-n = 0. We mainly investigate the refinement mask A(ξ) =1/|detM|∑_{α∈Z^d}a(α)e^{-iα·ξ} of the multivariatevector refinement equations satisfies the characterization of the eigenvalue conditionE whenξ= 0 , and as A(0) satisfies condition E we talk about the existence of the characterization of L₂-solutions , L_p- solutions and continuous solutions of the multivariate vector refinement equations.