Refinable subspaces of a refinable space

Abstract

Local refinable finitely generated shift-invariant spaces play a significant 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 begin with a refinable function $\psi :\mathbb {R}\to \mathbb {R}^{m}$ which is supported on $[0,1]$. We are interested in spaces generated by a function $\phi :\mathbb {R}\to \mathbb {R}^{n}$ built from the shifts of $\psi$.