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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Communicated by Robert S. Strichartz
Acknowledgements and Notes. The work was partially supported by a UGC Research Grant from CUHK.
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Fan, A.H., Lau, KS. Asymptotic behavior of multiperiodic functions\(G(x) = \prod\limits_{n = 1}^\infty {g(x/2^n )} \) . The Journal of Fourier Analysis and Applications 4, 129–150 (1998). https://doi.org/10.1007/BF02475985
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DOI: https://doi.org/10.1007/BF02475985