International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 3, Pages 139-148
doi:10.1155/S0161171202110209

On local properties of compactly supported solutions of the two-coefficient dilation equation

Abstract

Let a and b be reals. We consider the compactly supported solutions φ:ℝ→ℝ of the two-coefficient dilation equation φ(x)=aφ(2x)+bφ(2x−1). In this paper, we determine sets Ba,b, Ca,b, and Za,b defined in the following way: let x∈[0,1]. We say that x∈Ba,b (resp., x∈Ca,b, x∈Za,b) if the zero function is the only compactly supported solution of the two-coefficient dilation equation, which is bounded in a neighbourhood of x (resp., continuous at x, vanishes in a neighbourhood of x). We also give the structure of the general compactly supported solution of the two-coefficient dilation equation.