Equivalence of A-approximate continuity for self-adjoint expansive linear maps

摘要

Let A : R-d -> R-d, d >= 1, be an expansive linear map. 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): R-d -> R-d for which the respective concepts of A(mu)-approximate continuity (mu = 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. (C) 2008 Elsevier Inc. All rights reserved.