高维空间的框架多分辨分析与小波
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摘要
小波分析是近年来迅速发展起来的的一门应用数学学科，系统的研究开始于20世纪80年代初期。它从产生到现在虽然仅仅几十年的时间，但它在信号传输、图像处理、数字水印、偏微分方程、电磁场数值计算等众多领域都有广泛的应用。
本文主要从两个方面研究小波分析：一、从框架方面研究了近Riesz基的稳定性与摄动；二、从高维空间方面研究了框架多分辨分析的性质和小波。在第一章中，首先介绍了小波分析的发展历史及发展前景，然后介绍了本文的主要工作。在第二章中，先介绍了框架的定义与性质，接着描述了Riesz基、近Riesz基性质与它们各自之间的等价关系，最后重点研究了近Riesz基的稳定性与摄动问题，得出了关于近Riesz基摄动新的充要条件。在第三章中，在介绍了高维空间多分辨分析的定义与性质后，得出了高维空间框架多分辨分析的一些新的性质以及一个集合是框架多分辨分析的谱充要条件。在第四章中，我们首先介绍了高维空间的框架小波的定义与性质，然后介绍了小波集相关定义，得出了一个集合是相应小波集的等价条件。在第五章中，利用二维空间整数的分类，得出了的一个重要结论，在此基础上，得出二维框架多分辨分析存在一个Parseval框架小波的充要条件，并给出Parseval框架小波的表示形式。
Wavelet analyses is one of the disciplines of applied mathematics whichdeveloped rapidly in recent years, Research began in the early 80s of the 20thcentury. Ithas a wide range of applications such as signal processing, imageprocessing, digital watermarking, partial differential equations, numericalcalculation of electromagnetic fields and so on, though the wavelet analysesfrom produces to present merely several dozens years.
Two topics of the dissertation research about the wavelet analyses include:First, we studied the stability and the perturbation of the near-Riesz base fromframe; Second, we studied the properties of frame multiresolution analyses andwavelet in high-dimension spaces. In chapter 1, first introduced the developmenthistory and the prospects for development of the wavelet analyzes, then introducethe mainly conclusion of this article. In chapter 2, we first introduced theframe definition and properties, and then describes the properties of the Rieszbase and the near-Riesz base and their respective equivalence relations betweenthemseleves, finally we studied the stability and perturbation of the near-Rieszbase on emphasis, then obtained a new necessary and sufficient condition of theperturbation of the near-Riesz base. In chapter 3, first introduced thedefinition of frame multiresolution analyses in high-dimension space, thenobtained some properties and a necessary and sufficient condition of a set isa spectrum of frame multiresolution analyses. In chapter 4, we first introducedthe definition and properties of frame wavelet in high-dimension spaces, thenintroduced the definition of wavelet set, finally obtained a necessary andsufficient condition for a set is a wavelet set in a high-dimension spaces. Inchapter 5, we obtained a important conclusion though integer classification intwo-dimension space, use it, we obtaioned a necessary and sufficient conditionof multiresolutions analyses has the single Parseval frame wavelet in two- dimension space, and we give the express of Parseval frame wavelet.
本文主要从两个方面研究小波分析：一、从框架方面研究了近Riesz基的稳定性与摄动；二、从高维空间方面研究了框架多分辨分析的性质和小波。在第一章中，首先介绍了小波分析的发展历史及发展前景，然后介绍了本文的主要工作。在第二章中，先介绍了框架的定义与性质，接着描述了Riesz基、近Riesz基性质与它们各自之间的等价关系，最后重点研究了近Riesz基的稳定性与摄动问题，得出了关于近Riesz基摄动新的充要条件。在第三章中，在介绍了高维空间多分辨分析的定义与性质后，得出了高维空间框架多分辨分析的一些新的性质以及一个集合是框架多分辨分析的谱充要条件。在第四章中，我们首先介绍了高维空间的框架小波的定义与性质，然后介绍了小波集相关定义，得出了一个集合是相应小波集的等价条件。在第五章中，利用二维空间整数的分类，得出了的一个重要结论，在此基础上，得出二维框架多分辨分析存在一个Parseval框架小波的充要条件，并给出Parseval框架小波的表示形式。
Wavelet analyses is one of the disciplines of applied mathematics whichdeveloped rapidly in recent years, Research began in the early 80s of the 20thcentury. Ithas a wide range of applications such as signal processing, imageprocessing, digital watermarking, partial differential equations, numericalcalculation of electromagnetic fields and so on, though the wavelet analysesfrom produces to present merely several dozens years.
Two topics of the dissertation research about the wavelet analyses include:First, we studied the stability and the perturbation of the near-Riesz base fromframe; Second, we studied the properties of frame multiresolution analyses andwavelet in high-dimension spaces. In chapter 1, first introduced the developmenthistory and the prospects for development of the wavelet analyzes, then introducethe mainly conclusion of this article. In chapter 2, we first introduced theframe definition and properties, and then describes the properties of the Rieszbase and the near-Riesz base and their respective equivalence relations betweenthemseleves, finally we studied the stability and perturbation of the near-Rieszbase on emphasis, then obtained a new necessary and sufficient condition of theperturbation of the near-Riesz base. In chapter 3, first introduced thedefinition of frame multiresolution analyses in high-dimension space, thenobtained some properties and a necessary and sufficient condition of a set isa spectrum of frame multiresolution analyses. In chapter 4, we first introducedthe definition and properties of frame wavelet in high-dimension spaces, thenintroduced the definition of wavelet set, finally obtained a necessary andsufficient condition for a set is a wavelet set in a high-dimension spaces. Inchapter 5, we obtained a important conclusion though integer classification intwo-dimension space, use it, we obtaioned a necessary and sufficient conditionof multiresolutions analyses has the single Parseval frame wavelet in two- dimension space, and we give the express of Parseval frame wavelet.
引文
[1] Meyer Y. Wavelets and Operators[M]. Cambridge, UK. Cambridge University Press, 1992.
[2] Chui C.K. and Lian J. A Study of Orthonormal Multi-Wavelets[J]. Appl. Numer. Math. 1996,20(3): 273-298.
[3] Jiang Q. On the Design of Multifilter Banks and Orthonormal Multiwavelets Banks[J]. IEEE Trans. On SP.1998,46(12): 3292-3303.
[4] Goodman T. N. T. and Lee S. L. Wavelets of Multiplicity r. Trans[J]. Amen Math. Soc. 1994,342(1): 307-324.
[5] Xia X.G., Geronimo J.S., Hardin D.P. and et al. Design of Prefilters for Discrete Multi Wavelet Transforms[J]. IEEE Trans. on SP., 1996,1(1): 25-35.
[6] Frank Kurth and Michael Clausen. Filter Bank and M-Band Wavelet Packet Algorithms in Audio Signal Processing[J]. IEEE Trans. on SP.1999,47(2):549-554.
[7] 马鸿飞，樊昌信，宋国乡．基于M一频带小波变换的话宽带语音编码算法[J]．通信学报，1996．6，19(6)：20-25。
[8] Cohen A., Daubechies I. and Feauveau J.-C. Biorthogonal Bases of Compactly Supported Wavelets[J]. Comm. Pure Appl. Math, 1992,42: 485-560.
[9] Vetterli M. and Herley C. Wavelets and Filter Banks: Theory and Design[J]. IEEE Trans on SP.1998, 40(9): 2549-2554.
[10] Unser M., Therenaz P. and Aldroubi A. Shift-Othogonal Wavelet Bases[J]. IEEE on SP. 1998, 46(7): 1827-1836.
[12] 张贤达，保铮，非平稳信号分析与处理[M]．北京：国防工业出版社，1998．
[13] 王俊，陈逢时，张守宏．一种利用子波变换多分辨特性的信号消噪技术[J]．信号理，1996．6，12(2)105—109．
[14] S. Mallat Zero-Crossings of a Wavelet Transform[J]. IEEE Trans. on IT. 1991,37 (4):1019-1033.
[15] 陈永彬，王仁华．语言信号处理[M]．合肥：中国科学技术大学出版社，1990．5．
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[17] 成礼智，王红霞，罗永．小波的理论与应用[M]．北京：科学出版社，2004．
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[19] Peter G Casazza and Ole Christensen. Riesz frames and approximation of the frame coefficients [J], Appr. Theory and Appl. 1998, 14(2): 1-11.
[20] Peter G Casazza, Ole Christensen. Approximation of the frame coefficients using finite-dimensional method [J], Proc Amer Math Soc, 1998, 1(12): 99-112.
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[22] Peter G Casazza, Ole Christensen. Frames containing a Riesz basis and preservation of this property under perturbations [J], SIAM Math Anal. 1988, 29(1).1: 266-278.
[23] Peter G Casazza, Ole Christensen. Hilbert space frames containing a Riesz Basis and Banach spaces which have no subspace isomorphic to c_0 [J], Math Anal Appl, 1996, 202: 940-950.
[24] Peter G Casazz. Every frame is a sum of three (but not two) orthonormal bases-and other frame representations [J], Fourier Anal.Appl. 1988, 4(6): 727-732.
[25] Ole Christensen. A Paley-Winer Theorem for Frames [J], Proc Amer Math Soc, 1995, 123(7): 2199-2201.
[26] Ole Christensen. Frame, Riesz bases, and discerte Gabor/wavelet expansions [J], Bull Amer Math Soc, 2001, 38(3): 273-291.
[27] Ole Christensen. Frames and pseudo-inverse [J], Math Anal Appl, 1995, 191: 401-414.
[28] Ole Christensen. An Introduction to Frames and Riesz Bases [M], Birkhauser Boston, 2002: 87-115.
[29] Ole Christensen. Frames perturbation. [J], Proc Amer Math Soc, 1995, 123: 1217-1220.
[30] Ole Christensen. Frames in Hilbert space [D], Preprint 1995.
[31] Ole Christensen, C Heil. Perturbation of Banach frames and atomic decomposition [J], Math Nach. 1997, 185: 33-47.
[32] Ole Christensen, A Lindner. Frames. containing a Riesz basis and approximation of the inverse frame operator [J], Lin Aig and Its Appr Theory and Appl. 2001, 17(1): 117-130.
[33] Ole Christensen. Finite-dimensional approximation of the inverse frame operators and applications to Weyl-Heisenberg frames and wavelet frames [J], Fourier Anal.Appl.2000, (1): 79-91.
[34] Ole Christensen. Operators with closed range, pseudo-inverse, and perturbation of frames for a subspace [J], Canad Math Bull. 1999, 42(1): 37-45.
[35] Ole Christensen. Frames containing a Riesz basis and approximation of the frame coefficients using finite-dimensional methods [J], Math Anal Appl, 1996, 199: 256-270.
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[38] 辛杰，周家云：Banach空间框架和原子分解的摄动及稳定性[J]，数学学报2002，45(6)：1165-1170．
[39] 郭燕妮，周家云：Riesz框架的稳定性和摄动的研究[J]，数学学报 2003，46(4)：673-682．
[40] 郭燕妮，周家云Bessel框架以及摄动[J]，数学研究与评论 2003，23(3)：515-519．
[41] 周家云，刘宇，Banach空间框架和原子分解的性质[J]，数学学报，2004，47(3)：499-504．
[42] 丁友征，周家云，紧支撑小波框架的稳定性[J]，曲阜师范大学学报，2004，30(3)：267-288．
[43] 周家云，紧支撑Wey-Heisenberg框架的摄动[J]，系统科学与数学，2005，25(2)，170-178．
[44] 朱玉灿，Banach空间的框架与Riesz基[J]，应用数学，1988，11(4)：24-30．
[45] 朱玉灿，Banach空间的q-框架与p-Riesz基的稳定性[J]，数学年刊，22A：3(2001)17-25．
[46] W.Sun and X.Zhou, On the Stability of Multivariate Trigonometor Sytems [J], Math. Anal. Appl. 1999, 235: 159-167.
[47] 杨德运，周性伟，不规则小波框架的膨胀列，应用数学，2006，19(2)：421-425．
[48] Mallat S. Multiresolution approximations and wavelet orthonormal bases of L~2 (R) [J]. Amer. Math. Soc. 1989, 315: 69-87.
[49] Mallat S. Multiresolution represention and wavelete[J]. 1990, 11: 674-693.
[50] Daucheies.I, TenLectures onWavelet[M], Pliladephia: SIAM, 1992.
[51] Meyer Y．小波与算子(M)．北京：世界图书出版公司，1992，1-276．
[52] H.O.Kim, Lim J K. New characterizations of Riesz bases [J], Appl Comp Harm Anal, 1997, 4: 222-229.
[53] H.O.Kim, R.Y.Kim, On frame wavelets associated with frame multiresolution analyses [J], Appl. Comput. Hamon. Anal. 2001, 10: 61-70.
[54] H.O.Kim, R.Y.Kim, J.K.Lim, Sem-orthogonal frame wavelets and frame multi-resolution analyses[J], Bull. Austral. Math. SOC. 2002, 65: 35-44.
[55] H.O.Kim, R.Y.Kim, Quasi-biorthogonal frame multiresolution analyses and wavelets[J], Adv. Comput.Math.to appear. 2002, 12: 15-26.
[56] H.O.Kim, R.Y.Kim, On the Riesz wavelets associated with multiresolution analyses[J], Appl.Comput.Harmon.Anal. 2002, 13: 138-150.
[57] H.O.Kim, R.Y.Kim, On the spectrums of frame multiresolution analyses[J], Math.Appl. 2005, 305: 528-545.
[58] H.O.Kim, R.Y.Kim, J.K.Lim, Characterization of the closedness of the sum of two shift-invariant spaces[J], Math.App, 2005, 17(2): 1-15.
[59] Deyun Yang, Xingwei Zhou, Frame wavelets with matrix dilations in [J], Appl.Math. Letters. 2004, 17: 631-639.
[60] Mu Lehua, Zhang Zhihua, Zhang Peixun, On the higher-dimensional wavelet frames[J], Appl. Comput.Harmon.Anal. 2004, 16: 44-59.
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[70]Deyun Yang,Xingwei Zhou,Frame wavelets with matrix dilations in L~2(R~n) [J],Appl.Math 2004,17:631-639.
[71]Dorn Ervin Dutkay,Some equations relating multiwaveletes and multiscaling functions[J],Functions Aalysis.2005,226:1-20.
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[2] Chui C.K. and Lian J. A Study of Orthonormal Multi-Wavelets[J]. Appl. Numer. Math. 1996,20(3): 273-298.
[3] Jiang Q. On the Design of Multifilter Banks and Orthonormal Multiwavelets Banks[J]. IEEE Trans. On SP.1998,46(12): 3292-3303.
[4] Goodman T. N. T. and Lee S. L. Wavelets of Multiplicity r. Trans[J]. Amen Math. Soc. 1994,342(1): 307-324.
[5] Xia X.G., Geronimo J.S., Hardin D.P. and et al. Design of Prefilters for Discrete Multi Wavelet Transforms[J]. IEEE Trans. on SP., 1996,1(1): 25-35.
[6] Frank Kurth and Michael Clausen. Filter Bank and M-Band Wavelet Packet Algorithms in Audio Signal Processing[J]. IEEE Trans. on SP.1999,47(2):549-554.
[7] 马鸿飞，樊昌信，宋国乡．基于M一频带小波变换的话宽带语音编码算法[J]．通信学报，1996．6，19(6)：20-25。
[8] Cohen A., Daubechies I. and Feauveau J.-C. Biorthogonal Bases of Compactly Supported Wavelets[J]. Comm. Pure Appl. Math, 1992,42: 485-560.
[9] Vetterli M. and Herley C. Wavelets and Filter Banks: Theory and Design[J]. IEEE Trans on SP.1998, 40(9): 2549-2554.
[10] Unser M., Therenaz P. and Aldroubi A. Shift-Othogonal Wavelet Bases[J]. IEEE on SP. 1998, 46(7): 1827-1836.
[12] 张贤达，保铮，非平稳信号分析与处理[M]．北京：国防工业出版社，1998．
[13] 王俊，陈逢时，张守宏．一种利用子波变换多分辨特性的信号消噪技术[J]．信号理，1996．6，12(2)105—109．
[14] S. Mallat Zero-Crossings of a Wavelet Transform[J]. IEEE Trans. on IT. 1991,37 (4):1019-1033.
[15] 陈永彬，王仁华．语言信号处理[M]．合肥：中国科学技术大学出版社，1990．5．
[16] 李建平等．从傅立叶分析到小波分析：回顾与展[J]．计算机科学发，1999，26(12)： 29-30．
[17] 成礼智，王红霞，罗永．小波的理论与应用[M]．北京：科学出版社，2004．
[18] Aldroubi A. Portraits of frames [J], Proc Amer Math Soc, 1995. 123: 1661-1668.
[19] Peter G Casazza and Ole Christensen. Riesz frames and approximation of the frame coefficients [J], Appr. Theory and Appl. 1998, 14(2): 1-11.
[20] Peter G Casazza, Ole Christensen. Approximation of the frame coefficients using finite-dimensional method [J], Proc Amer Math Soc, 1998, 1(12): 99-112.
[21] Peter G Casazza, Ole Christensen. Perturbation of operators and applications to frame theory [J], Fourier Anal Appl. 1997, 3: 543-557.
[22] Peter G Casazza, Ole Christensen. Frames containing a Riesz basis and preservation of this property under perturbations [J], SIAM Math Anal. 1988, 29(1).1: 266-278.
[23] Peter G Casazza, Ole Christensen. Hilbert space frames containing a Riesz Basis and Banach spaces which have no subspace isomorphic to c_0 [J], Math Anal Appl, 1996, 202: 940-950.
[24] Peter G Casazz. Every frame is a sum of three (but not two) orthonormal bases-and other frame representations [J], Fourier Anal.Appl. 1988, 4(6): 727-732.
[25] Ole Christensen. A Paley-Winer Theorem for Frames [J], Proc Amer Math Soc, 1995, 123(7): 2199-2201.
[26] Ole Christensen. Frame, Riesz bases, and discerte Gabor/wavelet expansions [J], Bull Amer Math Soc, 2001, 38(3): 273-291.
[27] Ole Christensen. Frames and pseudo-inverse [J], Math Anal Appl, 1995, 191: 401-414.
[28] Ole Christensen. An Introduction to Frames and Riesz Bases [M], Birkhauser Boston, 2002: 87-115.
[29] Ole Christensen. Frames perturbation. [J], Proc Amer Math Soc, 1995, 123: 1217-1220.
[30] Ole Christensen. Frames in Hilbert space [D], Preprint 1995.
[31] Ole Christensen, C Heil. Perturbation of Banach frames and atomic decomposition [J], Math Nach. 1997, 185: 33-47.
[32] Ole Christensen, A Lindner. Frames. containing a Riesz basis and approximation of the inverse frame operator [J], Lin Aig and Its Appr Theory and Appl. 2001, 17(1): 117-130.
[33] Ole Christensen. Finite-dimensional approximation of the inverse frame operators and applications to Weyl-Heisenberg frames and wavelet frames [J], Fourier Anal.Appl.2000, (1): 79-91.
[34] Ole Christensen. Operators with closed range, pseudo-inverse, and perturbation of frames for a subspace [J], Canad Math Bull. 1999, 42(1): 37-45.
[35] Ole Christensen. Frames containing a Riesz basis and approximation of the frame coefficients using finite-dimensional methods [J], Math Anal Appl, 1996, 199: 256-270.
[36] Holub J K. Pre-framees, and near-Riesz bases in operators, Besselian fram Hilbert spaces. Proc. Amer. Math. Soc. 1994, 122(3): 779～785.
[37] Holub J K. Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces [J], Proc Amer Math Soc, 1994, 122(3): 77-78.
[38] 辛杰，周家云：Banach空间框架和原子分解的摄动及稳定性[J]，数学学报2002，45(6)：1165-1170．
[39] 郭燕妮，周家云：Riesz框架的稳定性和摄动的研究[J]，数学学报 2003，46(4)：673-682．
[40] 郭燕妮，周家云Bessel框架以及摄动[J]，数学研究与评论 2003，23(3)：515-519．
[41] 周家云，刘宇，Banach空间框架和原子分解的性质[J]，数学学报，2004，47(3)：499-504．
[42] 丁友征，周家云，紧支撑小波框架的稳定性[J]，曲阜师范大学学报，2004，30(3)：267-288．
[43] 周家云，紧支撑Wey-Heisenberg框架的摄动[J]，系统科学与数学，2005，25(2)，170-178．
[44] 朱玉灿，Banach空间的框架与Riesz基[J]，应用数学，1988，11(4)：24-30．
[45] 朱玉灿，Banach空间的q-框架与p-Riesz基的稳定性[J]，数学年刊，22A：3(2001)17-25．
[46] W.Sun and X.Zhou, On the Stability of Multivariate Trigonometor Sytems [J], Math. Anal. Appl. 1999, 235: 159-167.
[47] 杨德运，周性伟，不规则小波框架的膨胀列，应用数学，2006，19(2)：421-425．
[48] Mallat S. Multiresolution approximations and wavelet orthonormal bases of L~2 (R) [J]. Amer. Math. Soc. 1989, 315: 69-87.
[49] Mallat S. Multiresolution represention and wavelete[J]. 1990, 11: 674-693.
[50] Daucheies.I, TenLectures onWavelet[M], Pliladephia: SIAM, 1992.
[51] Meyer Y．小波与算子(M)．北京：世界图书出版公司，1992，1-276．
[52] H.O.Kim, Lim J K. New characterizations of Riesz bases [J], Appl Comp Harm Anal, 1997, 4: 222-229.
[53] H.O.Kim, R.Y.Kim, On frame wavelets associated with frame multiresolution analyses [J], Appl. Comput. Hamon. Anal. 2001, 10: 61-70.
[54] H.O.Kim, R.Y.Kim, J.K.Lim, Sem-orthogonal frame wavelets and frame multi-resolution analyses[J], Bull. Austral. Math. SOC. 2002, 65: 35-44.
[55] H.O.Kim, R.Y.Kim, Quasi-biorthogonal frame multiresolution analyses and wavelets[J], Adv. Comput.Math.to appear. 2002, 12: 15-26.
[56] H.O.Kim, R.Y.Kim, On the Riesz wavelets associated with multiresolution analyses[J], Appl.Comput.Harmon.Anal. 2002, 13: 138-150.
[57] H.O.Kim, R.Y.Kim, On the spectrums of frame multiresolution analyses[J], Math.Appl. 2005, 305: 528-545.
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