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

Letaandbbe reals. We consider the compactly supported solutionsφ:ℝ→ℝof the two-coefficient dilation equationφ(x)=aφ(2x)+bφ(2x−1). In this paper, we determine setsBa,b,Ca,b, andZa,bdefined in the following way: letx∈[0,1]. We say thatx∈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 ofx(resp., continuous atx, vanishes in a neighbourhood ofx). We also give the structure of the general compactly supported solution of the two-coefficient dilation equation.