中国物理B ›› 2008, Vol. 17 ›› Issue (6): 2018-2022.doi: 10.1088/1674-1056/17/6/014

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Fractional Fourier transform of Cantor sets: further numerical study

高穹, 廖天河, 崔远峰   

  1. Department of Mathematics and Physics, Information Engineering University, Zhengzhou 450001, China
  • 收稿日期:2007-10-15 修回日期:2007-11-08 出版日期:2008-06-20 发布日期:2008-06-20

Fractional Fourier transform of Cantor sets: further numerical study

Gao Qiong(高穹)†, Liao Tian-He(廖天河), and Cui Yuan-Feng(崔远峰)   

  1. Department of Mathematics and Physics, Information Engineering University, Zhengzhou 450001, China
  • Received:2007-10-15 Revised:2007-11-08 Online:2008-06-20 Published:2008-06-20

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摘要： This paper is a further work of the authors' paper published previously (Liao T H and Gao Q 2005 {\it Chin. Phys. Lett.} {\bf 22} 2316). The amplitudes of fractional Fourier transform of Cantor sets are analysed from the viewpoint of multifractal by wavelet transform maxima method (WTMM). An integral operation is carried out before the application of WTMM, such that the function obtained can be considered as the perturbed devil staircase. Also, wavelets with large number of vanishing moments are used, which makes the complete singularity spectrum more accessible. The validity of multifractal formalism is guaranteed by restricting parameter $q$ to a proper range, so that the phenomenon of multifractal phase transition can be explained reasonably. Particularly, the method of determining the range of parameter $q$ in the above paper is developed to be more operational and rigorous.

关键词: multifractal, fractional Fourier transform, wavelet transform, perturbed devil staircase

Abstract: This paper is a further work of the authors' paper published previously (Liao T H and Gao Q 2005 Chin. Phys. Lett. 22 2316). The amplitudes of fractional Fourier transform of Cantor sets are analysed from the viewpoint of multifractal by wavelet transform maxima method (WTMM). An integral operation is carried out before the application of WTMM, such that the function obtained can be considered as the perturbed devil staircase. Also, wavelets with large number of vanishing moments are used, which makes the complete singularity spectrum more accessible. The validity of multifractal formalism is guaranteed by restricting parameter $q$ to a proper range, so that the phenomenon of multifractal phase transition can be explained reasonably. Particularly, the method of determining the range of parameter $q$ in the above paper is developed to be more operational and rigorous.

Key words: multifractal, fractional Fourier transform, wavelet transform, perturbed devil staircase

中图分类号:  (Fourier analysis)

  • 02.30.Nw
02.10.Ab (Logic and set theory) 02.30.Uu (Integral transforms) 02.60.-x (Numerical approximation and analysis) 05.45.Df (Fractals) 05.70.Fh (Phase transitions: general studies)