Topological and Geometric Properties of Refinable Functions and Mra Affine Frames

We investigate some topological and geometric properties of the set R of all refinable functions in L(R), and of the set of all MRA affine frames. We prove that R is nowhere dense in L(R); the unit sphere of R is path-connected in the L-norm; and for any M -dimensional hyperplane generated by L-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 L(R). We also obtain a new characterization of the L-closure R of R, and extend the above topological and geometric results from R to R, and even further to the set of all refinable vectors and its L-closure.