On the existence of irregular solutions of the two-coefficient dilation equation

Summary.

We prove that for every real a and b such that $ab \neq 0$ there exists a very irregular compactly supported solution $\varphi : {\Bbb R} \rightarrow {\Bbb R}$ of the functional equation¶$\varphi(x)=a\varphi(2x)+b\varphi(2x-1)$.