Equivalence of A-Approximate Continuity for Self-Adjoint Expansive Linear Maps

Abstract

Let A be an expansive linear map from R^d to R^d. The notion of A-approximate continuity was recently used to give a characterization of scaling functions in a multiresolution analysis (MRA). The definition of A-approximate continuity at a point x - or, equivalently, the definition of the family of sets having x as point of A-density - depend on the expansive linear map A. The aim of the present paper is to characterize those self-adjoint expansive linear maps A_1, A_2 for which the respective concepts of A_j-approximate continuity (j=1,2) coincide. These we apply to analyze the equivalence among dilation matrices for a construction of systems of MRA. In particular, we give a full description for the equivalence class of the dyadic dilation matrix among all self-adjoint expansive maps. If the so-called ``four exponentials conjecture'' of algebraic number theory holds true, then a similar full description follows even for general self-adjoint expansive linear maps, too.