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On the Fourier Transforms of Nonlinear Self-similar Measures

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Abstract

In-homogeneous self-similar measures can be viewed as special cases of nonlinear self-similar measures. In this paper, we study the asymptotic behaviour of the Fourier transforms of nonlinear self-similar measures. Some typical examples are exhibited, and we show that the Fourier transforms of those measures are usually localized, i.e., the Fourier transforms decay rapidly at \(\infty \). We also discuss the infinity lower Fourier dimension of in-homogeneous self-similar measures and obtain its non-trivial bounds. The result confirms Conjecture 2.3 in Olsen and Snigireva (Math Proc Camb Philos Soc 144:465–493, 2008).

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Acknowledgements

This work was supported by the scientific research fund of the Education Department of Hunan Province under grant No. 19K019.

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Authors and Affiliations

  1. College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082, People’s Republic of China

    Zhanqi Zhang & Yingqing Xiao

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  1. Zhanqi Zhang
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  2. Yingqing Xiao
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Correspondence to Yingqing Xiao.

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Communicated by Dorin Dutkay.

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Zhang, Z., Xiao, Y. On the Fourier Transforms of Nonlinear Self-similar Measures. J Fourier Anal Appl 26, 43 (2020). https://doi.org/10.1007/s00041-020-09743-9

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  • DOI: https://doi.org/10.1007/s00041-020-09743-9

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