Refinement equations with nonnegative coefficients

In this paper we analyze solutions of the n-scale functional equation Φ(x) = Σk∈ℤpkΦ(nx - k), where n ≥ 2 is an integer, the coefficients pk are nonnegative and Σpk= 1. We construct a sharp criterion for the existence of absolutely continuous solutions of bounded variation. This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations and the problem of absolutely continuity of distribution function of one random series. Further we obtain a complete classification of refinement equations with positive coefficients (in the case of finitely many terms) with respect to the existence of continuous or integrable compactly supported solutions. © 2000 Birkhäuser Boston. All rights reserved.