In this paper we analyze solutions of the n-scale functional equation Î¦(x) = Î£kââ¤pkÎ¦(nx - k), where n â¥ 2 is an integer, the coefficients pk are nonnegative and Î£pk= 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. Â© 2000 BirkhÃ¤user Boston. All rights reserved.
|Titolo:||Refinement equations with nonnegative coefficients|
|Data di pubblicazione:||2000|
|Appare nelle tipologie:||1.1 Articolo in rivista|