Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W k p (R)(1 6 p 6 ∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W k p (R) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the Lp(1 6 p 6 ∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ ∈ Lp(R) (φ ∈ C(R) when p = ∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν, Lp(R)) for some ν > 0. This paper generalizes the results in [R. Q. Jia, K. S. Lau, and D. X. Zhou, J. Fourier Anal. Appl., 7 (2001), pp. 143–167] in the univariate setting to the multivariate setting.