Refinable subspaces of a refinable space

Local refinable finitely generated shift-invariant spaces play a significan t role in many areas of approximation theory and geometric design. In this paper we present a new approach to the construction of such spaces. We begi n with a refinable function psi : R --> R-m which is supported on [0; 1]. W e are interested in spaces generated by a function phi : R --> R-n built fr om the shifts of psi.