Asymptotic behavior of multiperiodic functionsG(x) = \prod\limits_{n = 1}^\infty {g(x/2^n )} |
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Authors: | Ai Hua Fan Ka-Sing Lau |
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Institution: | (1) Department of Mathematics, University Cergy-Pontoise, Cergy-Pontoise, France;(2) Department of Mathematics, The Chinese University of Hong Kong, Hong Kong;(3) Department of Mathematics, University of Pittsburgh, 15260 Pittsburgh, PA |
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Abstract: | Let 0≤g be a dyadic Hölder continuous function with period 1 and g(0)=1, and let $G(x) = \prod\nolimits_{n = 0}^\infty {g(x/{\text{2}}^n )} $ . In this article we investigate the asymptotic behavior of $\smallint _0^{\rm T} \left| {G(x)} \right|^q dx$ and $\frac{1}{n}\sum\nolimits_{k = 0}^n {\log g(2^k x)} $ using the dynamical system techniques: the pressure function and the variational principle. An algorithm to calculate the pressure is presented. The results are applied to study the regulatiry of wavelets and Bernoulli convolutions. |
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Keywords: | 41A63 41A65 28A80 |
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