Dilation equations and Markov operators (Q2567312)

Using the theory of Markov operators \(P: L^1(X)\to L^1(X)\) with \(P\) linear, \(\| Pf\|_1=\| f\|_1= 1\) for \(f\geq 0\), \(f\in L^1(X)\), the author gives a new proof for the following Theorem. Let \(N,k> 1\) be positive integers and \(c_0,\dots, c_N\geq 0\) be reals such that \(\sum^\infty_{i=0} c_{ki+j}= 1\) for every \(j\in \{0,\dots,k- 1\}\), where \(c_n= 0\) for all \(n> N\). Then the dilation equation \(f(x)= \sum^N_{n=0} c_n f(kx- n)\) has a nontrivial solution.