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Asymptotic behaviour of multiperiodic functions scaled by Pisot numbers

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Abstract

Let β > 1 be a Pisot number andg be a positive Hölder continuous function with period one and g(0) = 1. The multiperiodic functionG(ξ)=Π ∞n=0 g(ξ/βn) is studied and the asymptotic behaviour ofI G(T) = ∫ T0 G(ξ)dξ investigated. We prove that the limit of logI(T)/ logT exists asT tends to infinity. We also provide a method to calculate this limit for the caseg(ξ) = cos2 2πξ, corresponding to the Fourier transform of the Bernoulli convolution associated to the golden number (or some of its generalizations).

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Authors and Affiliations

  1. Department of Mathematics LAMFA CNRS FRE 2270, University of Picardie, 33 Rue Saint Leu, 80039, Amiens Cedex 1, France

    Ai Hua Fan

  2. Department of Mathematics, Wuhan University, 430072, Wuhan, China

    Ai Hua Fan

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  1. Ai Hua Fan
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Correspondence to Ai Hua Fan.

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Fan, A.H. Asymptotic behaviour of multiperiodic functions scaled by Pisot numbers. J. Anal. Math. 86, 271–287 (2002). https://doi.org/10.1007/BF02786652

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  • DOI: https://doi.org/10.1007/BF02786652

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