基于Sierpinski分形上的多分辨分析
A Multiresolution Analysis Based on Sierpinski Fractal
DOI: 10.12677/PM.2019.98124, PDF,   
作者: 李岱琪, 李万社：陕西师范大学，数学与信息科学学院，陕西 西安
关键词: Sierpinski垫；分形；多分辨分析；框架；Sierpinski Gasket； Fractal； Multiresolution Analysis； Framework
摘要: 根据经典的由迭代仿射函数系统产生的分形的扩张和平移所生成的σ有限测度空间上的多分辨分析的概念，本文讨论了基于Sierpinski垫上的多分辨分析，通过定义酉扩张算子D和酉算子T，并结合分形几何中的测度知识证明了基于Sierpinski垫的膨胀分形集上的多分辨分析。
Abstract: According to the classic theory, the notion of multiresolution analysis based on σ-finite measure spaces built from dilations and translations on a fractal arising from an iterated affine function system. A multiresolution analysis based on Sierpinski gasket is discussed in this paper. By defining unitary dilation operator D and unitary operator T, and combining the measure knowledge in fractal geometry, the multiresolution analysis on the inflated fractal set based on Sierpinski gasket is proved.
文章引用：李岱琪, 李万社. 基于Sierpinski分形上的多分辨分析[J]. 理论数学, 2019, 9(8): 980-988. https://doi.org/10.12677/PM.2019.98124

参考文献

[1] Yang, S.Z. and Cheng, Z.X. (2002) Orthonormal Multi-Wavelets on the Interval [0,1] with Multiplicity. Acta Mathematica Sinica, 45, 789-796.
[2] Nakahira, K. and Miyamoto, A. (2018) Parseval Wavelets on Hierarchical Graphs. Applied and Computational Harmonic Analysis, 44, 414-445. [Google Scholar] [CrossRef] 
[3] 师东利, 李万社. 基于二维康托尔二元群的小波变换[J]. 纺织高校基础科学学报, 2017, 30(3): 311-317.
[4] 聂伟平, 师东利, 李万社. 二维康托尔集上的多分辨分析[J]. 西安文理学院学报, 2016, 19(5): 26-29.
[5] 成礼智. 小波的理论与应用[M]. 北京: 科学出版社, 2004: 75-77.
[6] Dutkay, D. and Jorgensen, P.E.T. (2006) Wavelets on Fractals. Revista Matematica Iberoamericana, 22, 131-180. [Google Scholar] [CrossRef] 
[7] 方清城. MATLAB R2016a小波分析22个算法实现[M]. 北京: 电子工业出版社, 2018: 338-342.
[8] 沙震, 阮火军. 分形与拟合[M]. 杭州: 浙江大学出版社, 2005: 16-18.
[9] Christensen, O. (2005) Wavelet Frames and Multiresolution Analysis. Pure and Applied Mathematics Series, 272, 73-103. [Google Scholar] [CrossRef] 
[10] Bohnstengel, J. and Kessebohmer, M. (2010) Wavelets for Iterated Function Systems. Functional Analysis, 259, 583-601. [Google Scholar] [CrossRef] 
[11] David, G. and Semmes, S. (1997) Fractured Fractals and Broken Dreams. Clarendon Press, Oxford.
[12] D’Andrea, J., Merrill, K.D. and Packer (2008) Fractal Wavelets of Dutkay-Jorgensen Type for the Sierpinski Gasket Spaces. Communications in Contemporary Mathematics, 451, 69-88. [Google Scholar] [CrossRef] 
[13] Baggett, L.W., Medina, H.A. and Merrill, K.D. (1999) Generalized Multi-Resolution Analyses and a Construction Procedure for All Wavelet Sets in R^n. Fourier Analysis Applications, 5, 563-573. [Google Scholar] [CrossRef] 
[14] Strichartz, R.S. (1997) Piecewise Linear Wavelets on Sierpinski Gasket Type Fractals. The Journal of Fourier Analysis and Applications, 3, 387-415. [Google Scholar] [CrossRef] 
[15] Cao, S.P. and Qiu, H. (2017) Some Properties of the Derivatives on Sierpinski Gasket Type Fractals. Constructive Approximation, 46, 319-347. [Google Scholar] [CrossRef] 

为你推荐