Topological and geometric properties of refinable functions and MRA affine frames

Abstract

We investigate some topological and geometric properties of the set R of all refinable functions in L2(Rd), and of the set of all MRA affine frames. We prove that R is nowhere dense in L2(Rd); the unit sphere of R is path-connected in the L2-norm; and for any M-dimensional hyperplane generated by L2-functions f0,...,fM, either almost all the functions in the hyperplane are refinable or almost all the functions in the hyperplane are not refinable. We show that the set of all MRA affine frames is nowhere dense in L2(Rd). We also obtain a new characterization of the L2-closure R̄ of R, and extend the above topological and geometric results from R to R̄, and even further to the set of all refinable vectors and its L2-closure. © 2010 Elsevier Inc. All rights reserved.