Refinement equations with nonnegative coefficients

  • Vladimir Protasov


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

15A24 26C10 30B20 42A32 42A38 

Keywords and Phrases

refinement equation Fourier series tree distributions joint spectral radius 


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Copyright information

© Birkhäuser 2000

Authors and Affiliations

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

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