Dilation equations and Markov operators ✩

Using the theory of Markov operators, we give a new proof of the known fact saying that for every positive integers N andk >1, and for every nonnegative reals c0, . . . , cN satisfying the first sum rule the dilation equation f (x) = N n=0 cnf (kx −n) has a nontrivial L1-solution.  2005 Elsevier Inc. All rights reserved