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

• Published in: Linear algebra and its applications Vol. 429; no. 7; pp. 1504 - 1521
• Main Authors: REVESZ, Sz. Gy, SAN ANTOLIN, A
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01.10.2008; Elsevier Science
• Subjects: [formula omitted]-approximate continuity; Algebra; Exact sciences and technology; Linear and multilinear algebra, matrix theory; Mathematics; Multiresolution analysis; Point of [formula omitted]-density; Sciences and techniques of general use; Self-adjoint expansive linear map; Self-adjoint expansive linear map; Point of A -density; 15A99; 15A21; 26B35; A -approximate continuity; 15A36; Multiresolution analysis; Algebraic number theory; Equivalence classes; Dilatation; A-approximate continuity; Multiresolution analysis; Point of A-density; Self-adjoint expansive linear map; Continuous function; 26B35; 15A21; 15A36; 15A99; Self adjoint operator; Canonical form; Integer matrix
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2008.04.028

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 μ -approximate continuity ( μ = 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.