Abstract
In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σk∈ℤ Pk 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.
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Communicated by Albert Cohen
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Protasov, V. Refinement equations with nonnegative coefficients. The Journal of Fourier Analysis and Applications 6, 55–78 (2000). https://doi.org/10.1007/BF02510118
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DOI: https://doi.org/10.1007/BF02510118