MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP

doi:10.1016/S0252-9602(09)60102-8

数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (5): 1251-1266.doi: 10.1016/S0252-9602(09)60102-8

MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP

刘和平, 刘宇, 王海辉

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, |China Department of Mathematics and Mechanics, School of Applied Science, University of Science and Technology Beijing, Beijing 100083, China Department of Applied Mathematics, Beihang University, Beijing 100083, China

收稿日期:2007-08-16 出版日期:2009-09-20 发布日期:2009-09-20
基金资助:
Sponsored by the NSFC (10871003, 10701008, 10726064), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007001040)

MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP

LIU He-Ping, LIU Yu, WANG Hai-Hui

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, |China Department of Mathematics and Mechanics, School of Applied Science, University of Science and Technology Beijing, Beijing 100083, China Department of Applied Mathematics, Beihang University, Beijing 100083, China

Received:2007-08-16 Online:2009-09-20 Published:2009-09-20
Supported by:
Sponsored by the NSFC (10871003, 10701008, 10726064), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007001040)

摘要/Abstract

摘要：

In this article, the properties of multiresolution analysis and self-similar tilings on the Heisenberg group are
studied. Moreover, we establish a theory to construct an orthonormal Haar wavelet base in $L^2({\mathbb H}^d)$ by using self-similar tilings for the acceptable dilations on the Heisenberg group.

关键词: Heisenberg group, multiresolution analysis, wavelets, self similar tilings

Abstract:

Key words: Heisenberg group, multiresolution analysis, wavelets, self similar tilings

中图分类号:

43A15

刘和平, 刘宇, 王海辉. MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP[J]. 数学物理学报(英文版), 2009, 29(5): 1251-1266.

LIU He-Ping, LIU Yu, WANG Hai-Hui. MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP[J]. Acta mathematica scientia,Series B, 2009, 29(5): 1251-1266.

参考文献

[1] Folland G B. Harmonic Analysis in Phase Space. New Jersey: Princeton University Press, 1989

[2] Franchi B. Rectifiability and perimeter in the Heisenberg group. Math Ann, 2001, 321: 479--531

[3] Gelbrich G. Self-similar periodic tilings on the Heisenberg group. J Lie Theory, 1994, 4: 31--37

[4] Gröchenig K, Madych W R. Multiresolution analysis, Haar bases, and self-similar tilings of Rⁿ. IEEE Transactions on Information Theory, 1992, 38: 556--568

[5] Jawerth B, Peng L Z. Compactly supported orthogonal wavelets on the Heisenberg group. Beijing: Research Report No.45 of Insititute Mathematics of Peking University, 2001

[6] Kigami J. Analysis on Fractals. Cambridge: Cambridge University Press, 2001

[7] Lawton W. Infinite convolution products & refinable distributions on Lie groups. Trans Amer Math Soc, 2000, 352: 2913--2936

[8] Liu H P, Peng L Z. Admissible wavelets associated with the Heisenberg group. Pacific J Math, 1997, 180: 101--123

[9] Liu H P, Liu Y, Peng L Z, et al. Cascade algorithm and multiresolution analysis on the Heisenberg
group. Progress in Natural Science, 2005, 15: 602--608

[10] Liu H P, Liu Y. Refinable functions on the Heisenberg group. Comm Pure Appl Anal, 2007, 6: 775--787

[11] Liu M J, Lu S Z. A weighted estimate of the Hormander multiplier on the Heisenberg group. Acta Math Sci, 2006, 26B(1): 134--144

[12] Liu Y, Yu M. Lipschitz continuity of refinable functions on the Heisenberg group. J Math Anal Appl, 2008, 338: 1081--1091

[13] Peng L Z. Wavelets on the Heisenberg Group. Geometry and Nonlinear Partial Differential Equations, AMS/IP Studies in Advanced Mathematics, 2002, 29: 123--131

[14] Stein E M. Harmonic Analysis Real-Variable Methods, Orthogonality, and Oscillatory Integrals. New Jersey: Princeton University Press, 1993

[15] Strichartz R S. Self-similarity on nilpotent Lie group. Contemporary Mathematics, 1992, 140: 123--157

[16] Strichartz R S. Wavelet and self-affine tilings. Constr Approx, 1993, 9: 327--346

[17] Yang Q. D. Multiresolution analysis on non-abelian locally compact groups
[D]. University of Saskatchewan, 1999

MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP

MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP

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