A probabilistic approach to scaling equations (Q1818258)

The paper deals with compactly supported and measurable solutions \(f:R\to R\) of the dilation equation \[ f(x)=\sum_{n=0}^{N}c_nf(2x-n), \tag{1} \] where the \(c_n\)'s are real coefficients. Although some general results on equation (1) are presented, the paper concerns the three-coefficient case of (1); it is the three-coefficient dilation equation \[ f(x)=c_0f(2x)+c_1f(2x-1)+c_2f(2x-2). \tag{2} \] Assuming \(c_0+c_1+c_2=2\) the authors get characterizations of all the coefficients \(c_0\) and \(c_1\) for which equation (2) has a compactly supported and measurable solution \(f\) such that \(f(x)+f(x+1)=1\) a.e. in \([0,1]\). Moreover, explicit representations of all such solutions are obtained. Under the same assumption \(c_0+c_1+c_2=2\) necessary and sufficient conditions on the coefficients \(c_0\) and \(c_1\) for which equation (2) has a unique \(L^p\) solution \(f\) with compact support and such that \(f(x)+f(x+1)=1\) a.e. in \([0,1]\) are given.