Journal of Fourier Analysis and Applications

, Volume 6, Issue 1, pp 55–78

Refinement equations with nonnegative coefficients

  • Vladimir Protasov

DOI: 10.1007/BF02510118

Cite this article as:
Protasov, V. The Journal of Fourier Analysis and Applications (2000) 6: 55. doi:10.1007/BF02510118


In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σk∈ℤPk Ф(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.

Math Subjec Classifications


Keywords and Phrases

refinement equationFourier seriestreedistributionsjoint spectral radius

Copyright information

© Birkhäuser 2000

Authors and Affiliations

  • Vladimir Protasov
    • 1
  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia