抽样定理与Parseval框架小波的若干问题研究
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摘要
自1986年以来,小波分析已成为科学研究中的热点领域,其应用涉及自然科学与工程技术的许多领域。目前,小波分析已成为研究和解决自然科学与工程计算中许多复杂问题的强有力工具。在小波分析受到广泛应用的信号处理和通信领域中,抽样定理起着举足轻重的作用,它建立了离散信号和模拟信号的等价关系。小波子空间中抽样定理的研究是当今国际上抽样定理发展的一个重要方向。作为正交小波基的推广,Parseval框架小波保持了正交小波基除正交性以外的几乎所有的良好性质。如今,框架理论已经被成功地应用在信号处理、图像处理、数据压缩、抽样理论等领域,因此研究Parseval框架小波具有非常重要的意义。
本文分为两个部分。第一部分研究了Parseval框架小波与Parseval框架尺度函数的构造和性质;第二部分研究了L2 ( R )子空间中非整点抽样定理和基于框架的多小波子空间的抽样定理。主要结果概括为以下几个方面:
1.综述了小波分析产生的背景及小波理论和小波子空间中抽样定理若干重要方面的研究现状,并介绍了本文的主要工作。
2.考虑到通过广义多分辨分析(GMRA)构造的小波都具有半正交性,因此,我们首先研究矩阵伸缩半正交框架小波的性质,给出其成立的必要条件,证明具有附加条件的半正交框架小波成立的充要条件。其次,发现所有GMRA Parseval框架小波与{T kψ: k∈Zd}为一闭子空间W0的Parseval框架的等价性,从而进一步完善了广义多分辨分析的理论基础。最后,通过小波的最小向量滤波器刻画了GMRA Parseval框架小波的一个重要性质。
3.研究了Parseval框架尺度函数集与MSF Parseval框架小波,得出许多有意义的结论。首先给出可测集S为Parseval框架尺度函数集的充要条件。其次,通过研究可测集S的性质,得到一种构造Parseval框架尺度函数集的方法,并证明所有的Parseval框架尺度函数集都由这种方法构成。最后,依靠揭示MSF Parseval框架小波与Parseval框架尺度函数集之间的关系,我们得到两个结果:一是每个Parseval框架尺度函数集生成一个满足一定条件的MSF Parseval框架小波ψ;二是对于维数函数几乎处处满足Dψ(ξ) =χ? (ξ)≤1的MSF Parseval框架小波ψ,一定存在一个Parseval框架尺度函数集S使得ψ? =χBS \S成立。由于MSF Parseval框架小波ψ可以由它的Fourier变换ψ? =χW来定义,研究小波的性质就可以简化为研究可测集W的性质。因此,我们的结果不但提供了一种构造MSF Parseval框架小波的新方法,而且也加深了人们对小波的理解。
4.把文献[101]的结果推广到Parseval框架尺度函数的情况,得到可测集G为Parseval框架尺度函数的频域支集的充要条件。特别是通过讨论集合G、τ( G)、12G和G \12G之间的关系,揭示了基于多分辨分析的尺度函数与基于Parseval框架多分辨分析的尺度函数之间的不同。同时,通过研究有限带宽Parseval框架小波的频域支集,得到它的一个重要性质。
5.讨论了L2 ( R )子空间中基于框架的非整点抽样定理。我们依靠平移算子和伸缩算子性质来证明( )L2 R子空间中a > 0步长非整点抽样定理成立的充要条件,同时也给出了其抽样函数在频域中的表示形式。
6.研究了多小波子空间中非整点规则抽样与非规则抽样。首先,通过Zak变换提供一种构造满足一定条件的非整点规则抽样函数的方法,同时建立了基于框架的非整点规则抽样定理。其次,注意到非规则抽样具有很强的实用价值,因此,我们也研究了与框架相关的非规则抽样定理,得到其成立的必要条件。然后,基于所得结果,讨论非整点规则抽样的扰动问题,给出计算扰动的方法。
最后,对整个论文的工作和研究成果进行了总结,讨论了一些有待于进一步研究的问题以及下一步的研究设想和目标。
Since 1980's, wavelet analysis has been a popular field in scientific research. Its application almost involves all the branches in natural science and engineering technology. Nowadays, it has been a powerful tool for exploring and solving many complicated problems in natural science and engineering computation. Wavelet analysis is applied widely in signal processing and digital communications, and the sampling theorem plays a crucial role in signal processing and digital communications too: it tells us how to convert an analog sinal into a sequence of numbers. The sampling theorem in the wavelet subspaces is a popular field in modern stage. As the generalization of orthogonal bases, Parseval frame wavelet almost preserve the nice properties of orthogonal bases except the orthogonality. So far, frame theory is successfully used by many fields such as signal processing, mapping processing, data compress and sampling theorem. Thus, it is important to study Parseval frame wavelet.
The dissertation is composed of two parts. In part one, we discuss the construction and property of Parseval frame wavelets and Parseval frame scaling functions. In part two, we investigate the lattic sampling theorem in subspace of L2 ( R ) and sampling theorem associated to frame in multiwavelet subspace. The important and significant results obtained in this dissertation can be sumarized as follows:
1.We outline the development histories of wavelet analysis and some current research situations of wavelet theory and the sampling theorem in the wavelet subspaces. Main results of the thesis will be presented.
2.Notice that all the waveltes constructed by general multiresolution analysis(GMRA) have the property of sime-orthogonal, so we firstly study the property of semi-orthogonal frame wavelet, and get the necessary conditions for the semi-orthogonal frame wavelet to hold, and prove the necessary and sufficent conditions for the semi-orthogonal frame wavelet to hold. Then, we discover that all the GMRA Parseval wavelets are equivalent to the affine system {T kψ: k∈Zd} being the Parseval frame of a closed subspace W0 such that improving the theoretical basis of GMRA. Finally, by the minimal vector-filter, we depict an important property of frame wavelets.
3.We consider the Parseval frame scaling function set and MSF Parseval frame wavelets, and get several significant results. At first, we show the necessary and sufficent conditions for a measurable set S to be the Parseval frame scaling function set. Then, by discussing the property of the measurable set S, we provide a method of construction of Parseval frame scaling function set, and prove that all the Parseval frame scaling function sets are obtained by the construction method. At last, by discovering the relation between the MSF Parseval frame wavelet and the Parseval frame scaling function set, we get two important results: Firstly, each MSF Parseval frame wavelet with the additional property arise from the Parseval frame scaling function set. Secondly, for the dimention function of MSF Parseval frame waveletψsatisfies Dψ(ξ) =χ? (ξ)≤1, there must exist a Parseval frame scaling function set s such thatψ? =χBS \S. By the MSF Parseval frame wavelet defined by its Fourier transformψ? =χW, the study of the property of the wavelet is reduced to the study of the property of the set W. So, we not only provide a new construction method of the MSF Parseval frame wavelet, but also have a better understanding of wavelets in general.
4.We generalize the results in [101] to the Parseval frame scaling function, and derive the necessary and sufficent conditions for the measurable set G to be the support G of the Fourier transform of the Parseval frame scaling functions. In the special case, by studying the relations among the sets G,τ( G)、12G and G \12G , we discover the difference between the MRA scaling function and the FMRA scaling function. And, by considering the support of the Fourier transform of the band-limited scaling function, we get an important property of the band-limited scaling function.
5.We discuss the lattice sampling theorem based on frame in subspace of L2 ( R ). Through investigating the property of the translation operator and the dilation operator, we prove a necessary and sufficient condition for a > 0 lattice sampling theorem to hold in subspace of L2 ( R ), and obtain the formula of lattice sampling function in frequency domain.
6 . We study the uniform noninteger sampling theorem and irregular sampling in multiwavelet subspace. Firstly, by Zak transform, we provide a construction method for the uniform noninteger sampling function with the additional property, and establish the uniform noninteger sampling theorem based on frame in multiwavelet subspace. Then, by irregular sampling be also useful in practice, we consider the irregular sampling theorem based on frame, and obtain the necessary condition for the irregular sampling theorem to hold in multiwavelet subspace. At last, from our obtained results, we discuss the perturbations of uniform noninteger sampling, and establish the algorithm for perturbations of uniform noninteger sampling in multiwavelet subspace.
Finally , sums up the work and research results, brings forward future research considerations and objects.
本文分为两个部分。第一部分研究了Parseval框架小波与Parseval框架尺度函数的构造和性质;第二部分研究了L2 ( R )子空间中非整点抽样定理和基于框架的多小波子空间的抽样定理。主要结果概括为以下几个方面:
1.综述了小波分析产生的背景及小波理论和小波子空间中抽样定理若干重要方面的研究现状,并介绍了本文的主要工作。
2.考虑到通过广义多分辨分析(GMRA)构造的小波都具有半正交性,因此,我们首先研究矩阵伸缩半正交框架小波的性质,给出其成立的必要条件,证明具有附加条件的半正交框架小波成立的充要条件。其次,发现所有GMRA Parseval框架小波与{T kψ: k∈Zd}为一闭子空间W0的Parseval框架的等价性,从而进一步完善了广义多分辨分析的理论基础。最后,通过小波的最小向量滤波器刻画了GMRA Parseval框架小波的一个重要性质。
3.研究了Parseval框架尺度函数集与MSF Parseval框架小波,得出许多有意义的结论。首先给出可测集S为Parseval框架尺度函数集的充要条件。其次,通过研究可测集S的性质,得到一种构造Parseval框架尺度函数集的方法,并证明所有的Parseval框架尺度函数集都由这种方法构成。最后,依靠揭示MSF Parseval框架小波与Parseval框架尺度函数集之间的关系,我们得到两个结果:一是每个Parseval框架尺度函数集生成一个满足一定条件的MSF Parseval框架小波ψ;二是对于维数函数几乎处处满足Dψ(ξ) =χ? (ξ)≤1的MSF Parseval框架小波ψ,一定存在一个Parseval框架尺度函数集S使得ψ? =χBS \S成立。由于MSF Parseval框架小波ψ可以由它的Fourier变换ψ? =χW来定义,研究小波的性质就可以简化为研究可测集W的性质。因此,我们的结果不但提供了一种构造MSF Parseval框架小波的新方法,而且也加深了人们对小波的理解。
4.把文献[101]的结果推广到Parseval框架尺度函数的情况,得到可测集G为Parseval框架尺度函数的频域支集的充要条件。特别是通过讨论集合G、τ( G)、12G和G \12G之间的关系,揭示了基于多分辨分析的尺度函数与基于Parseval框架多分辨分析的尺度函数之间的不同。同时,通过研究有限带宽Parseval框架小波的频域支集,得到它的一个重要性质。
5.讨论了L2 ( R )子空间中基于框架的非整点抽样定理。我们依靠平移算子和伸缩算子性质来证明( )L2 R子空间中a > 0步长非整点抽样定理成立的充要条件,同时也给出了其抽样函数在频域中的表示形式。
6.研究了多小波子空间中非整点规则抽样与非规则抽样。首先,通过Zak变换提供一种构造满足一定条件的非整点规则抽样函数的方法,同时建立了基于框架的非整点规则抽样定理。其次,注意到非规则抽样具有很强的实用价值,因此,我们也研究了与框架相关的非规则抽样定理,得到其成立的必要条件。然后,基于所得结果,讨论非整点规则抽样的扰动问题,给出计算扰动的方法。
最后,对整个论文的工作和研究成果进行了总结,讨论了一些有待于进一步研究的问题以及下一步的研究设想和目标。
Since 1980's, wavelet analysis has been a popular field in scientific research. Its application almost involves all the branches in natural science and engineering technology. Nowadays, it has been a powerful tool for exploring and solving many complicated problems in natural science and engineering computation. Wavelet analysis is applied widely in signal processing and digital communications, and the sampling theorem plays a crucial role in signal processing and digital communications too: it tells us how to convert an analog sinal into a sequence of numbers. The sampling theorem in the wavelet subspaces is a popular field in modern stage. As the generalization of orthogonal bases, Parseval frame wavelet almost preserve the nice properties of orthogonal bases except the orthogonality. So far, frame theory is successfully used by many fields such as signal processing, mapping processing, data compress and sampling theorem. Thus, it is important to study Parseval frame wavelet.
The dissertation is composed of two parts. In part one, we discuss the construction and property of Parseval frame wavelets and Parseval frame scaling functions. In part two, we investigate the lattic sampling theorem in subspace of L2 ( R ) and sampling theorem associated to frame in multiwavelet subspace. The important and significant results obtained in this dissertation can be sumarized as follows:
1.We outline the development histories of wavelet analysis and some current research situations of wavelet theory and the sampling theorem in the wavelet subspaces. Main results of the thesis will be presented.
2.Notice that all the waveltes constructed by general multiresolution analysis(GMRA) have the property of sime-orthogonal, so we firstly study the property of semi-orthogonal frame wavelet, and get the necessary conditions for the semi-orthogonal frame wavelet to hold, and prove the necessary and sufficent conditions for the semi-orthogonal frame wavelet to hold. Then, we discover that all the GMRA Parseval wavelets are equivalent to the affine system {T kψ: k∈Zd} being the Parseval frame of a closed subspace W0 such that improving the theoretical basis of GMRA. Finally, by the minimal vector-filter, we depict an important property of frame wavelets.
3.We consider the Parseval frame scaling function set and MSF Parseval frame wavelets, and get several significant results. At first, we show the necessary and sufficent conditions for a measurable set S to be the Parseval frame scaling function set. Then, by discussing the property of the measurable set S, we provide a method of construction of Parseval frame scaling function set, and prove that all the Parseval frame scaling function sets are obtained by the construction method. At last, by discovering the relation between the MSF Parseval frame wavelet and the Parseval frame scaling function set, we get two important results: Firstly, each MSF Parseval frame wavelet with the additional property arise from the Parseval frame scaling function set. Secondly, for the dimention function of MSF Parseval frame waveletψsatisfies Dψ(ξ) =χ? (ξ)≤1, there must exist a Parseval frame scaling function set s such thatψ? =χBS \S. By the MSF Parseval frame wavelet defined by its Fourier transformψ? =χW, the study of the property of the wavelet is reduced to the study of the property of the set W. So, we not only provide a new construction method of the MSF Parseval frame wavelet, but also have a better understanding of wavelets in general.
4.We generalize the results in [101] to the Parseval frame scaling function, and derive the necessary and sufficent conditions for the measurable set G to be the support G of the Fourier transform of the Parseval frame scaling functions. In the special case, by studying the relations among the sets G,τ( G)、12G and G \12G , we discover the difference between the MRA scaling function and the FMRA scaling function. And, by considering the support of the Fourier transform of the band-limited scaling function, we get an important property of the band-limited scaling function.
5.We discuss the lattice sampling theorem based on frame in subspace of L2 ( R ). Through investigating the property of the translation operator and the dilation operator, we prove a necessary and sufficient condition for a > 0 lattice sampling theorem to hold in subspace of L2 ( R ), and obtain the formula of lattice sampling function in frequency domain.
6 . We study the uniform noninteger sampling theorem and irregular sampling in multiwavelet subspace. Firstly, by Zak transform, we provide a construction method for the uniform noninteger sampling function with the additional property, and establish the uniform noninteger sampling theorem based on frame in multiwavelet subspace. Then, by irregular sampling be also useful in practice, we consider the irregular sampling theorem based on frame, and obtain the necessary condition for the irregular sampling theorem to hold in multiwavelet subspace. At last, from our obtained results, we discuss the perturbations of uniform noninteger sampling, and establish the algorithm for perturbations of uniform noninteger sampling in multiwavelet subspace.
Finally , sums up the work and research results, brings forward future research considerations and objects.
引文
[1] Daubechies I. Ten Lectures on Wavelets[M]. Philadelphia: SIAM Publ, 1992.
[2] Daubechies I. Orthonormal bases of compactly supported wavelets[J]. Comm Pure Appl Math. 1988,41(7),909-996.
[3] Daubechies I, Groddmann A, Mayer Y. Painless nonorthogonal expansions [J]. J Math Phys, 1986, 27(5): 1271-1283.
[4]迈耶著.小波与算子[M].尤众译,邓东皋,龙瑞麟校.北京:世界图书出版公司, 1992.
[5]程正兴.小波分析算法与应用[M].西安:西安交通大学出版社, 1997.
[6] Guo K, Labate D, Lim W Q, Weiss G. Wavelets with composite dilations and their MRA properties[J]. Appl Comput Harmon Anal, 2006, 20(2): 202-236.
[7]冯象初,甘小冰,宋国乡.数值泛函与小波理论[M].西安:西安电子科技大学出版, 2003.
[8]李建平,唐远炎.小波分析方法的应用[M].重庆:重庆大学出版社, 1999.
[9]刘贵忠,邸双亮.小波分析及其应用[M].西安:西安电子科技大学出版社, 1992.
[10] Str?mberg J O. A modified Franklin system and higher order spline systems on R n as unconditional bases for Hardy space.ProcConfin honor of Antomi Zygmund[C]. Wadsworth Mathseries, 1982, 2: 475-493.
[11] Battle G. A block spin construction of ondelettes, Part I: Lemariéfunctions[J]. Comm Math Phys, 1987, 110(4): 601-615.
[12] LemariéP G. Une nouvelle base ondeletts de L2 (R n)[J]. De Math Pureset Appl, 1988, 315(9), 69-88.
[13] Mallat S. A theory for multiresolution signal decomposition: the wavelet representation[J]. IEEE Trans Pattern Anal Machine Intell, 1989, 11(7): 674-693.
[14] Chui C K, Wang J Z. On compactly supported spline wavelets and a duality principle[J]. Trans Amer Math Soc, 1992, 330(2): 903-915.
[15] Cohen A. Ondelettes analyses multirésolutions et filters miroir en quadrature[J]. Ann Inst H PoincaréAnalnonlinéaire, 1990, 7(5): 439-459.
[16] Mallat S. A wavelet tour of signal proceesing[M]. San Diego: Academic Press, 1998.
[17] Coifman R, Meyer Y, Wickerhauser M V. Size properties of wavelets[A]. Wavelets and Their Applications[C], Boston, 1992, 153-178.
[18] Lawton W. Necessary and sufficient for constructing orthonormal wavelet bases[J]. Math Phys, 1991, 32(1): 57-61.
[19] Lewis R M. Cardinal interpolation multiresolutions[J]. Approx Theory, 1994, 26(2): 177-202.
[20] Cohen A, Daubechies I, Feauveau J C. Biorthogonal bases of compactly supportedwavelets[J]. Comm Pure Appl Math, 1992, 45(5): 485-560.
[21] Sweldens W. The lifting scheme: A custom-design constructionof biorthogonal wa- velets[J]. Appl Comput Harmon Anal, 1996, 3(2): 186-200.
[22] Sweldens W. The lifting scheme:A construction of second generation of wavelets[J]. SIAM Math Anal, 1997, 29(2): 511-546.
[23] Daubechies I, Sweldens W. Factoring wavelet transforms into lifting steps[J]. Fourier Anal Appl. 1998, 4(3): 245-267.
[24]黄达人,毕宁.多进制小波分析[M].杭州:浙江大学出版社, 2001.
[25] Chui C K, Lian J. Construction of compactly supported symmetric and antisymmetric orthonormal wavelets with scale=3[J]. Appl Comput Harmon Anal, 1995, 2(1): 21-51.
[26] Han B. Symmetric orthonormal scaling functions and wavelets with dilation factor 4[J]. Adv Comput Math, 1998, 8(3): 221-247.
[27] Steffen P, Heller P N, Gopinath R A, Burrus C S. Theory of regular M-band wavelet bases[J]. IEEE Trans Signal Processing, 1993, 41(12): 3497-3511.
[28] Heller P. Rank M wavelets with N vanishing moments[J]. SIAM Matrix Anal Appl- ication, 1994, 16(2): 502-519.
[29] Soardi P M. Biorthogonal M-channel compactly supported wavelets [J]. Constr App- rox, 2000, 16(2): 283-311.
[30] Maggioni M. M-band Burt-Adelson biorthogonal wavelts[J]. Appl Comput Harm Anal, 2000, 9(3): 286-311.
[31] Resnikoff H L, Tian J, Wells R O Jr. Biorthogonal wavelet space: parametrization and factorization[J]. SIAM Math Anal, 2001, 33(1): 194-215.
[32] Bi N, Dai X, Sun Q. Construction of compactly supported M-band wavelets[J]. Appl Comput Harmon Anal, 1999, 6(2): 113-131.
[33] Ji H, Shen Z. Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions[J]. Adv Comput Math, 1999, 11(1): 81-104.
[34] Shui P L, Bao Z, Zhang X D. M-band compactly supported orthogonal symmetric interpolating scaling functions[J]. IEEE Trans Signal Processing, 2001, 49(8): 1704-1713.
[35]毕宁,黄达人,戴青云,李峰.一类含参数的正交、对称四进小波的构造[J].计算数学, 2005, 27(2): 141-150.
[36]张泽银.一类具有对称性和插值性的紧支撑尺度函数[J].数学学报, 2003, 46(3): 347-350.
[37] Goodman T N T, Lee S L. Wavelets of multiplicity r[J]. Trans Amer Math Soc, 1994, 342(1): 307-324.
[38] Donovan G C, Geronimo J, Hardin D P, Massopust P. Contruction of orthogonal wavelets using fractal interpolation functions[J]. SIAM Math Anal, 1996, 27(4): 1158-1192.
[39] Chui C K, Lian J. A study of orthonormal multiwavelets[J]. App Numer Math, 1996, 20(3): 273-298.
[40] Strela V. Multiwavelet:Theory and Applications[D]. PhD, Thesis MIT, 1996.
[41] Plonka G, Strela V. Construction of multiscaling function with approximation and symmetry[J]. SIAM Math Anal, 1998, 29(2): 172-199.
[42] Yang S Z, Cheng Z X, Wang H Y. Construction of biorthogonal multiwavelets[J]. Math Anal Appl, 2002, 276(1): 1-12.
[43]杨守志,唐远炎,程正兴. a尺度紧支撑正交多小波的构造[J].计算数学, 2002, 24(4): 451-460.
[44]杨守志.小波的构造、算法及其应用[D].西安:西安交通大学, 2001.
[45] Jiang Q. Orthogonal multiwavelets with optimum time-frequency resolution[J]. IEEE Trans Signal Processing, 1998, 46(4): 830-845.
[46] Jiang Q. On the design of multifilter banks and orthonormal multiwavelets bases[J]. IEEE Trans Signal Processing, 1998, 46(12): 3292-3302.
[47] Jiang Q. Symmetric paraunitary matrix extension and parametrization of symmetrization of symmetric orthogonal multifilter banks[J]. SIAM Matrix Anal Appl, 2001, 23(1): 167-186.
[48] Han B, Jiang Q. Multiwavelets on the interval[J]. Appl Comput Harmon Anal, 2002, 12(1): 100-127.
[49] Wilson R G, Meyer F G, Coifman R R. Adaptive wavelet packet basis selection for zerotree image coding[J]. IEEE Trans Image Processing, 2003, 12(12): 1460-1472.
[50] Lian J, Chui C K. Balanced multiwavelets with short filters[J]. IEEE Signal Processing Letters. 2004, 11(2): 75-78.
[51] Xia X G, Suter B W. Vector-valued wavelets and vector filter banks[J]. IEEE Trans Signal Processing, 1996, 44(3): 508-518.
[52] Bacchelli S, Cotroonei M, Sauer T. Wavelets for multichannel signals[J]. Adv Appl Math, 2002, 29(4): 581-598.
[53] Fowler J E, Li H. Wavelet Transforms for Vector Fields Using Omnidirectionally Balanced Multiwavelets[J]. IEEE Trans Signal Processing, 2002, 50(12): 3018-3027.
[54] Rieder P, G?tze J, Nossek J A. Implementation of discrete vector-valued wavelet transforms[R]. Hong kong: IEEE International Symposium on Circuits and Systems, 1997.
[55] He W, Lai M J. Examples of bivariate nonseparable compactly supported continuous wavelets[J]. IEEE Trans Image Processing, 2000, 9(5): 949-953.
[56] Riemenschneider S D, Shen Z. Construction of compactly supported biorthogonal wavelets[C]. In I N E Mastorakis, physisics and modern topics in mechanical and electrical engineering, Singapore: World Scientific, 1999, 201-206 Available at http://www. math. nus. edu. sg/ matzuows/ publist.html.
[57] Chen D, Han B, Riemenschneider S D. Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments[J]. Adv Comput Math. 2000, 13(2): 131-165.
[58] Ji H, Riemenschneider S D, Shen Z. Multivariate compactly supported fundamental refinables functions, duals and biorthogonalwavelets[J]. Studiesin Appl Math, 1999, 102(2): 173-204.
[59] Han B. Analysis and construction of optimal multivariate biorthogonal wavelets with compact support[J]. SIAM Math Anal, 1999, 30(2): 274-304.
[60] Logar A, Sturmfels B. Algorithm for the Quillen-Suslin theorem[J]. Journal of Algeba, 1992, 145(1): 231-239.
[61] Ron A, Shen Z. Affine systems in L2 (R d): the analysis of the analysis operator[J]. Funct Anal, 1997, 148(2): 408-447.
[62] Benedetto J J, Li S. The theory of multiresolution analysis frames and applications to filter banks[J]. Appl Comput Harmon Anal, 1998, 5(4): 389-427.
[63] Chui C K, He W. Compactly supported tight frames associated with refinable func-tions[J]. Appl Comput Harmon Anal, 2000, 8(3): 293-319.
[64] Brut P J, Adelon E H.The Laplacian pyramid as a compact image code[J].IEEE Trans Commun,1983,31(4): 532-540.
[65] Petukhov A. Explicit construction of Framelets[J]. Appl Comput Harmon Anal, 2001, 11(2): 313-327.
[66] Daubechies I, Grossmann A, Meyer Y. Painless nonorthogonal expansions[J]. J Math Phys, 1986, 27(5): 1271-1283.
[67] Li S. A theory of generalized multiresolution structure and pseudoframes of trans-lates[J]. Fourier Anal Appl, 2001, 7(1): 23-40.
[68] Sun W, Zhou X. Density and stability of wavelet frames[J]. Appl Comput Harm Anal, 2003, 15(2): 117-133.
[69] Chui C K, He W, Stockler J. Nonstationary tight wavelet frames, I: Bounded intervals[J]. Appl Comput Harmon Anal, 2004, 17(2): 141-197.
[70] Xu M. Wavelets frame asociated to accretive function[J]. Math Anal Appl, 2004, 295(1): 153-162.
[71] Casazza P G. The art of frame theory[J]. Taiwanese Journal of Mathematics, 2000, 4(2): 129-201.
[72] Christensen O. An Introduction to Frames and Riesz bases[M]. Birkh?user, Boston, MA, 2003.
[73] Li S, Ogawa H. Pseudoframes for Subspaces with Applications[J]. Fourier Anal Appl, 2004, 10(4): 409-431.
[74] Salvatori M, Soardi P M. Multivariate tight affine frames with a small number of generators[J]. Appro Theory, 2004, 127(1): 61-73.
[75] Christensen O, Eldar Y C. Oblique dual frames and shift-invariant spaces[J]. Appl Comput Harmon Anal, 2004, 17(1): 48-68.
[76] Weber E. Orthogonal frames of translates[J]. Appl Comput Harmon Anal, 2004, 17(1): 69-90.
[77] Czaja W, Kutyniok G, Speegle D. The geometry of sets of parameters of wave packet frames[J]. Appl Comput Harmon Anal, 2006, 20 (1): 108-125.
[78] Gabardo J P, Han D. The uniqueness of the dual of Weyl-Heisenberg subspace frames[J]. Appl Comput Harmon Anal, 2004, 17(2): 226-240.
[79] Labate D, Wilson E. Connectivity in the set of Gabor frames[J]. Appl Comput Harmon Anal, 2005, 18(1): 123-136.
[80] Benedetto J J, Pfander G E. Frame expansions for Gabor multipliers[J]. Appl Comput Harmon Anal, 2006, 20(1): 26-40.
[81]李松.在(L p (R s ))r(1≤p≤∞)中的向量细分格式[J].中国科学(A缉), 2003, 33(1): 1-11.
[82] Jia R, Jiang Q, Shen Z. Convergence of cascade algorithm associated with nonhomogeneous refinement equations[J]. Proc Amer Math Soc, 2001(2), 129: 415-427.
[83] Micchelli C A, Sauer T. On vector subdivision[J]. Math Z, 1998, 229: 621-674.
[84] Jia R, Riemenschneider S D, Zhou D. Vector subdivision schemes and multiple wavelets[J]. Mathematics of Computation, 1998, 67(224): 1533-1563.
[85]李云章,田雄飞.一类二维不可分非正交小波包.应用数学学报[J]. 2001, 24(3): 418-424.
[86] Chen Q, Cheng Z. A study on compactly supported orthogonal vector-valued wavelets and wavelet packets[J]. Chaos, Solitons & Fractals, 2007, 31(4): 1024-1034.
[87] Chang C S, Jin J. Separation of corona using wavelet packet transform and neural network for detection of partial discharge in gasinsulated substations[J]. IEEE Trans Power Delivery, 2005, 20(2): 1363-1369.
[88] He H, Cheng S. Home network power-line communication signal processing based on wavelet packet analysis[J]. IEEE Trans Power Delivery, 2005, 20(3): 1879-1885.
[89] Chui C K, Quak E. Wavelets on a bounded interval[R]. CAT Report # 265, Texas: Texas A & M University, 1992.
[90] Micchelli C A, Sauer T. Regularity of multiwavelets[J]. Adv Comput Math, 1997, 7(4): 455-545.
[91] Micchelli C A, Xu Y. Reconstruction and decomposition algorithms for biorthogonal multiwavelets[J]. Multidimensional Systems and Signal Process, 1997, 8(1): 31-69.
[92]崔丽鸿.紧支撑的小波、多小波理论及构造[D].西安:西安交通大学, 2004.
[93]关履泰.在[0,1]区间的自由结点样条小波[J].工程数学学报, 1998, 15(3): 1-6..
[94] Baggett L, Medina H, Merrill K. Generalized multiresolution analysis, and aconstruction procedure for all wavelet sets in R d[J]. Fourier Anal Appl, 1999, 5(6): 563-573.
[95] Walter G G. A sampling theorem for wavelet subspaces[J]. IEEE Trans Inform Theory, 1992, 38(2), 881-884.
[96] Bownik M. The structure of shift-invariant subspace of L2 ( R d)[J]. Jour Funct Anal, 2000, 177(2): 282-309.
[97] Baki? D. Semi-orthogonal Parseval frame wavelet and generalized multiresolution analyses[J]. Appl Comput Harmon Anal, 2006, 21(3): 281-304.
[98] Dai X. Diao Y, Gu Q, Han D. Frame wavelet sets in R d[J]. Jour Comput Appl Math, 2003, 155(1): 69-82.
[99] Walter G G. Translation and dilation invariance in orthogonal wavelet[J]. Appl Comput Harmon Anal, 1994, 1(4): 344-349.
[100] Walter G G. Wavelet and Other Orthogonal Systems with Applications[M]. Boca Raton, FL: CRC Press, 1994.
[101] Zhang Z. Supports of Fourier transforms of scaling functions[J]. Appl Comput Harmon Anal, 2007, 22(2): 141-156.
[102] Zhou X, Sun W. On the Sampling Theorem for Wavelet Subspaces[J]. Jour Fourier Anal Appl, 1999, 5(4): 347-354.
[103] Ron A, Shen Z. Frames and stable bases for shift-invariant subspaces of L2 ( R d)[J]. Canad J Math, 1995, 47(5): 1051-1094.
[104] Sun W, Zhou X. Sampling theorems for wavelet subspaces: error estimate and irregular sampling[J]. IEEE Trans Signal Process, 2000, 48(1): 223-226.
[105] Chen W, Itoh S, Shiki J. A sampling theorem for shift-invariant spaces[J]. IEEE Trans. Signal Processing, 1998, 46(10): 2802-2810.
[106] Zhao P, Liu G Z, Zhao C. Frames and Sampling Theorems for Translation-Invariant Subspaces[J]. IEEE Trans Signal Processing, 2004, 11(1): 8-11.
[107]龙瑞麟.高维小波分析[M].北京:世界图书出版公司, 1995.
[108] W Li, Zhang Y Q. A Study on the optimal attributes of transform domain vector quantization for image and video compression[R]. ICC, 1993.
[109] Sun L, Cheng Z X. Construction of a class of compactly supported orthogonal vector-valued wavelets[J]. Chaos, Solitons & Fractals, 2007, 34(2): 253-261.
[110] Shannon C E. Communication in the prensence of noise[J]. Proc Institute of Radio Engineers, 1949, 37 (1): 10–21.
[111] Sun W. Sampling theorems for multivariate shift invariant subspaces[J]. Sampl Theory Signal Image Process, 2005, 4(1): 73-98.
[112] Blanco C, Cabrelli C, Heineken S. Functions in Sampling Spaces[J], Sampl Theory Signal Image Process, 2005, 5(3): 275-265.
[113] Hernández E, Weiss G. A First Course on Wavelets[M]. Boca Raton, FL: CRC Press, 1996.
[114] Fornaiser M, Gori L. Sampling theorems on bounded domains[J]. Journal of Computational and Applied Mathematics, 2008, 221(2): 376-385.
[115] Zhao J, Tao R, Wang Y. Sampling rate conversion for linear canonical transform[J]. Signal Processing, 2008, 88(11): 2825-2832.
[116] Chen Q, Wang Y B, Wang Y. A sampling theorem for non-bandlimited signals using generalized Sinc functions[J]. Computer & Mathematics with Applications, 2008, 56(6): 1650-1661.
[117] Annaby M H, Bustoz J, Ismail M E H. On sampling theory and basic Sturm-Liouville systems[J]. Jounral of Computational and Applied Mathematics, 2007, 206(1): 73-85.
[118] Fenoy M M, Ibarrola P. The optional sampling theorem for submartingales in the sequentially planned context[J]. Statistics & Probability Letters, 2007, 77(8): 826-831.
[119] Benedetto J J, Sumetkijakan S. Tight frames and geometric properties of wavelet sets [J]. Adv Comput Math, 2006, 24(1): 35-56.
[120] Ron A, Shen Z. Weyl-Heisenberg frames and Riesz bases in L2 (R d)[J]. Duke Math J, 1997, 89(2): 237-282.
[121] Baki? D, Krishtal I, Wilson E N. Parseval frame wavelets with E n(2)-dilations[J]. Appl Comput Harmon Anal, 2005, 19(3): 386-431.
[122] Kim H O, Kim R Y, Lim J K. On the spectrum of frame multiresolution analysis[J]. J Math Anal Appl, 2005, 305 (2): 528-545.
[123] Dai X, Larson D. Wandering Vectors for Unitary Systems and Orthogonal Wavelets[J]. Memoirs of AMS, 1998, 136(651): No. 640,.
[124] Dai X, Diao Y, Gu Q. Subspaces with Normalized Tight Frame Wavelets in L2 (R )[J]. Proceedings of AMS, 2002, 130(6):1661-1667.
[125] Dai X, Diao Y, Gu Q, Han D. The existence of subspace wavelet sets[J]. Jounral of Computational and Applied Mathematics, 2003, 155(1): 83-90.
[126] Welland G V (Ed). Beyond Wavelets[M]. San Diego, CA: Academic Press , 2003.
[127] Candès E J, Donoho D L. Ridgelets: a key to higher-dimensional intermittency?[J]. Philos Trans Roy Soc, 1999, 357(1760): 2495-2509.
[128] Coifman R R, Meyer F G. Brushlets: A tool for directional image analysis and image compression[J]. Appl Comput Harmon Anal, 1997, 4(2): 147-187.
[129] Candès E J, Donoho D L. New tight frames of curvelets and optimal representations of objects with C 2 singularities[J]. Comm Pure Appl Math, 2004, 57(2): 219-266.
[130] Xia X G, Zhang Z. On sampling theorem , wavelet and wavelet transforms[J]. IEEE Trans Signal Processing, 1993, 41(12): 3524-3535.
[131] Janssen A J E M. The Zak transform and sampling theorems for wavelet subspaces[J]. IEEE Trans Signal Processing, 1993, 41(12): 3360-3365.
[132] Liu Y. Irregular sampling for spline wavelet subspaces[J]. IEEE Trans Inform Theory, 1996, 42(2): 623-627.
[133] Liu Y, Walter G G. Irregular sampling in wavelet subspaces[J]. Jour Fourier Anal Appl, 1995, 2(2): 181-189.
[134] Ojokovic I, Vaidyanatha P P. Generalized sampling theorems in multiresolution subspaces[J]. IEEE Trans Signal Processing, 1997, 45(3): 583-599.
[135] Chen W, Itoh S, Shiki J. Irregular sampling theorems for wavelet subspace[J]. IEEE Trans Inform Theory, 1998, 44(3): 1131-1142.
[136] Young R. An introduction to nonharmonic Fourier series[M]. New York: Academic Press, 1980.
[137] Walter G G. Wavelet subspaces with oversampling property[J]. Indag Math, 1993, 4(4): 499-507.
[138] Zayed A I, Benedetto J J, Paulo J S G, Eds. Shannon-type wavelets and the convergence of their associated wavelet series[A]. In Modern sampling theory[M], Boston: Brikh?user, 2001: 135-152.
[139] Li W. Vector transform and image coding[J]. IEEE Trans Circuits Syst Video Technol, 1991, 1(4): 297-307.
[140] Li W, Zhang Y Q. A Study on the optimal attributes of transform domain vector quantization for image and video compression[R]. ICC, 1993.
[141] Sun L, Cheng Z X. Construction of a class of compactly supported orthogonal vector-valued wavelets[J]. Chaos, Solitons & Fractals, 2007, 34(2): 253-261.
[142] Zhao P, Liu G, Zhao C. Frames and sampling theorems for translation-invariant subspaces[J]. IEEE Signal Process Lett, 2004, 11(1): 8–11.
[143] Aldroubi A, Grochenig K. Non-uniform sampling in shift-in-variant spaces[J]. SIAM Rev, 2001, 43(4): 585–620.
[144] Sun W, Zhou X. Frames and sampling theorem[J]. Sci China, 1998, 41(2): 606–612.
[145] Zhao P, Zhao C, Casazza P G. Perturbation of Regular Sampling in Shift-Invariant Spaces for Frames[J]. IEEE Trans Inf Theory, 2006 52(10): 4643–4648.
[146] Geronimo J S, Hardin D P, Massopust P R. Fractal functions and wavelet expansions based on several scaling functions[J]. J Approx Theory, 1994, 78(3): 373–401.
[147] Gao X. Notes on sampling theorem for multiwavelet[A]. in Proc IEEE ICSP[C], 2002, vol 1, Aug 26–30, 155–157.
[148] Selesnick I W. Interpolating multiwavelet bases and the sampling theorem[J]. IEEE Trans Signal Processing, 1999, 47(6): 1615–1621.
[149] Zhou D. Interpolatory orthogonal multiwavelets and refinable functions[J]. IEEE Trans Signal Processing, 2002, 50(3): 520–527.
[150] Zhao C, Zhao P. Sampling Theorem and Irregular Sampling Theorem for Multiwavelet Subspaces[J]. IEEE Trans Signal Processing, 2005, 53(2): 705–713.
[151] Benedetto J J, Li S. Multiresolution analysis frames with subspaces[A]. In Proc ICASSP 3[C], Minneapolis MN, 1993: 304-307.
[2] Daubechies I. Orthonormal bases of compactly supported wavelets[J]. Comm Pure Appl Math. 1988,41(7),909-996.
[3] Daubechies I, Groddmann A, Mayer Y. Painless nonorthogonal expansions [J]. J Math Phys, 1986, 27(5): 1271-1283.
[4]迈耶著.小波与算子[M].尤众译,邓东皋,龙瑞麟校.北京:世界图书出版公司, 1992.
[5]程正兴.小波分析算法与应用[M].西安:西安交通大学出版社, 1997.
[6] Guo K, Labate D, Lim W Q, Weiss G. Wavelets with composite dilations and their MRA properties[J]. Appl Comput Harmon Anal, 2006, 20(2): 202-236.
[7]冯象初,甘小冰,宋国乡.数值泛函与小波理论[M].西安:西安电子科技大学出版, 2003.
[8]李建平,唐远炎.小波分析方法的应用[M].重庆:重庆大学出版社, 1999.
[9]刘贵忠,邸双亮.小波分析及其应用[M].西安:西安电子科技大学出版社, 1992.
[10] Str?mberg J O. A modified Franklin system and higher order spline systems on R n as unconditional bases for Hardy space.ProcConfin honor of Antomi Zygmund[C]. Wadsworth Mathseries, 1982, 2: 475-493.
[11] Battle G. A block spin construction of ondelettes, Part I: Lemariéfunctions[J]. Comm Math Phys, 1987, 110(4): 601-615.
[12] LemariéP G. Une nouvelle base ondeletts de L2 (R n)[J]. De Math Pureset Appl, 1988, 315(9), 69-88.
[13] Mallat S. A theory for multiresolution signal decomposition: the wavelet representation[J]. IEEE Trans Pattern Anal Machine Intell, 1989, 11(7): 674-693.
[14] Chui C K, Wang J Z. On compactly supported spline wavelets and a duality principle[J]. Trans Amer Math Soc, 1992, 330(2): 903-915.
[15] Cohen A. Ondelettes analyses multirésolutions et filters miroir en quadrature[J]. Ann Inst H PoincaréAnalnonlinéaire, 1990, 7(5): 439-459.
[16] Mallat S. A wavelet tour of signal proceesing[M]. San Diego: Academic Press, 1998.
[17] Coifman R, Meyer Y, Wickerhauser M V. Size properties of wavelets[A]. Wavelets and Their Applications[C], Boston, 1992, 153-178.
[18] Lawton W. Necessary and sufficient for constructing orthonormal wavelet bases[J]. Math Phys, 1991, 32(1): 57-61.
[19] Lewis R M. Cardinal interpolation multiresolutions[J]. Approx Theory, 1994, 26(2): 177-202.
[20] Cohen A, Daubechies I, Feauveau J C. Biorthogonal bases of compactly supportedwavelets[J]. Comm Pure Appl Math, 1992, 45(5): 485-560.
[21] Sweldens W. The lifting scheme: A custom-design constructionof biorthogonal wa- velets[J]. Appl Comput Harmon Anal, 1996, 3(2): 186-200.
[22] Sweldens W. The lifting scheme:A construction of second generation of wavelets[J]. SIAM Math Anal, 1997, 29(2): 511-546.
[23] Daubechies I, Sweldens W. Factoring wavelet transforms into lifting steps[J]. Fourier Anal Appl. 1998, 4(3): 245-267.
[24]黄达人,毕宁.多进制小波分析[M].杭州:浙江大学出版社, 2001.
[25] Chui C K, Lian J. Construction of compactly supported symmetric and antisymmetric orthonormal wavelets with scale=3[J]. Appl Comput Harmon Anal, 1995, 2(1): 21-51.
[26] Han B. Symmetric orthonormal scaling functions and wavelets with dilation factor 4[J]. Adv Comput Math, 1998, 8(3): 221-247.
[27] Steffen P, Heller P N, Gopinath R A, Burrus C S. Theory of regular M-band wavelet bases[J]. IEEE Trans Signal Processing, 1993, 41(12): 3497-3511.
[28] Heller P. Rank M wavelets with N vanishing moments[J]. SIAM Matrix Anal Appl- ication, 1994, 16(2): 502-519.
[29] Soardi P M. Biorthogonal M-channel compactly supported wavelets [J]. Constr App- rox, 2000, 16(2): 283-311.
[30] Maggioni M. M-band Burt-Adelson biorthogonal wavelts[J]. Appl Comput Harm Anal, 2000, 9(3): 286-311.
[31] Resnikoff H L, Tian J, Wells R O Jr. Biorthogonal wavelet space: parametrization and factorization[J]. SIAM Math Anal, 2001, 33(1): 194-215.
[32] Bi N, Dai X, Sun Q. Construction of compactly supported M-band wavelets[J]. Appl Comput Harmon Anal, 1999, 6(2): 113-131.
[33] Ji H, Shen Z. Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions[J]. Adv Comput Math, 1999, 11(1): 81-104.
[34] Shui P L, Bao Z, Zhang X D. M-band compactly supported orthogonal symmetric interpolating scaling functions[J]. IEEE Trans Signal Processing, 2001, 49(8): 1704-1713.
[35]毕宁,黄达人,戴青云,李峰.一类含参数的正交、对称四进小波的构造[J].计算数学, 2005, 27(2): 141-150.
[36]张泽银.一类具有对称性和插值性的紧支撑尺度函数[J].数学学报, 2003, 46(3): 347-350.
[37] Goodman T N T, Lee S L. Wavelets of multiplicity r[J]. Trans Amer Math Soc, 1994, 342(1): 307-324.
[38] Donovan G C, Geronimo J, Hardin D P, Massopust P. Contruction of orthogonal wavelets using fractal interpolation functions[J]. SIAM Math Anal, 1996, 27(4): 1158-1192.
[39] Chui C K, Lian J. A study of orthonormal multiwavelets[J]. App Numer Math, 1996, 20(3): 273-298.
[40] Strela V. Multiwavelet:Theory and Applications[D]. PhD, Thesis MIT, 1996.
[41] Plonka G, Strela V. Construction of multiscaling function with approximation and symmetry[J]. SIAM Math Anal, 1998, 29(2): 172-199.
[42] Yang S Z, Cheng Z X, Wang H Y. Construction of biorthogonal multiwavelets[J]. Math Anal Appl, 2002, 276(1): 1-12.
[43]杨守志,唐远炎,程正兴. a尺度紧支撑正交多小波的构造[J].计算数学, 2002, 24(4): 451-460.
[44]杨守志.小波的构造、算法及其应用[D].西安:西安交通大学, 2001.
[45] Jiang Q. Orthogonal multiwavelets with optimum time-frequency resolution[J]. IEEE Trans Signal Processing, 1998, 46(4): 830-845.
[46] Jiang Q. On the design of multifilter banks and orthonormal multiwavelets bases[J]. IEEE Trans Signal Processing, 1998, 46(12): 3292-3302.
[47] Jiang Q. Symmetric paraunitary matrix extension and parametrization of symmetrization of symmetric orthogonal multifilter banks[J]. SIAM Matrix Anal Appl, 2001, 23(1): 167-186.
[48] Han B, Jiang Q. Multiwavelets on the interval[J]. Appl Comput Harmon Anal, 2002, 12(1): 100-127.
[49] Wilson R G, Meyer F G, Coifman R R. Adaptive wavelet packet basis selection for zerotree image coding[J]. IEEE Trans Image Processing, 2003, 12(12): 1460-1472.
[50] Lian J, Chui C K. Balanced multiwavelets with short filters[J]. IEEE Signal Processing Letters. 2004, 11(2): 75-78.
[51] Xia X G, Suter B W. Vector-valued wavelets and vector filter banks[J]. IEEE Trans Signal Processing, 1996, 44(3): 508-518.
[52] Bacchelli S, Cotroonei M, Sauer T. Wavelets for multichannel signals[J]. Adv Appl Math, 2002, 29(4): 581-598.
[53] Fowler J E, Li H. Wavelet Transforms for Vector Fields Using Omnidirectionally Balanced Multiwavelets[J]. IEEE Trans Signal Processing, 2002, 50(12): 3018-3027.
[54] Rieder P, G?tze J, Nossek J A. Implementation of discrete vector-valued wavelet transforms[R]. Hong kong: IEEE International Symposium on Circuits and Systems, 1997.
[55] He W, Lai M J. Examples of bivariate nonseparable compactly supported continuous wavelets[J]. IEEE Trans Image Processing, 2000, 9(5): 949-953.
[56] Riemenschneider S D, Shen Z. Construction of compactly supported biorthogonal wavelets[C]. In I N E Mastorakis, physisics and modern topics in mechanical and electrical engineering, Singapore: World Scientific, 1999, 201-206 Available at http://www. math. nus. edu. sg/ matzuows/ publist.html.
[57] Chen D, Han B, Riemenschneider S D. Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments[J]. Adv Comput Math. 2000, 13(2): 131-165.
[58] Ji H, Riemenschneider S D, Shen Z. Multivariate compactly supported fundamental refinables functions, duals and biorthogonalwavelets[J]. Studiesin Appl Math, 1999, 102(2): 173-204.
[59] Han B. Analysis and construction of optimal multivariate biorthogonal wavelets with compact support[J]. SIAM Math Anal, 1999, 30(2): 274-304.
[60] Logar A, Sturmfels B. Algorithm for the Quillen-Suslin theorem[J]. Journal of Algeba, 1992, 145(1): 231-239.
[61] Ron A, Shen Z. Affine systems in L2 (R d): the analysis of the analysis operator[J]. Funct Anal, 1997, 148(2): 408-447.
[62] Benedetto J J, Li S. The theory of multiresolution analysis frames and applications to filter banks[J]. Appl Comput Harmon Anal, 1998, 5(4): 389-427.
[63] Chui C K, He W. Compactly supported tight frames associated with refinable func-tions[J]. Appl Comput Harmon Anal, 2000, 8(3): 293-319.
[64] Brut P J, Adelon E H.The Laplacian pyramid as a compact image code[J].IEEE Trans Commun,1983,31(4): 532-540.
[65] Petukhov A. Explicit construction of Framelets[J]. Appl Comput Harmon Anal, 2001, 11(2): 313-327.
[66] Daubechies I, Grossmann A, Meyer Y. Painless nonorthogonal expansions[J]. J Math Phys, 1986, 27(5): 1271-1283.
[67] Li S. A theory of generalized multiresolution structure and pseudoframes of trans-lates[J]. Fourier Anal Appl, 2001, 7(1): 23-40.
[68] Sun W, Zhou X. Density and stability of wavelet frames[J]. Appl Comput Harm Anal, 2003, 15(2): 117-133.
[69] Chui C K, He W, Stockler J. Nonstationary tight wavelet frames, I: Bounded intervals[J]. Appl Comput Harmon Anal, 2004, 17(2): 141-197.
[70] Xu M. Wavelets frame asociated to accretive function[J]. Math Anal Appl, 2004, 295(1): 153-162.
[71] Casazza P G. The art of frame theory[J]. Taiwanese Journal of Mathematics, 2000, 4(2): 129-201.
[72] Christensen O. An Introduction to Frames and Riesz bases[M]. Birkh?user, Boston, MA, 2003.
[73] Li S, Ogawa H. Pseudoframes for Subspaces with Applications[J]. Fourier Anal Appl, 2004, 10(4): 409-431.
[74] Salvatori M, Soardi P M. Multivariate tight affine frames with a small number of generators[J]. Appro Theory, 2004, 127(1): 61-73.
[75] Christensen O, Eldar Y C. Oblique dual frames and shift-invariant spaces[J]. Appl Comput Harmon Anal, 2004, 17(1): 48-68.
[76] Weber E. Orthogonal frames of translates[J]. Appl Comput Harmon Anal, 2004, 17(1): 69-90.
[77] Czaja W, Kutyniok G, Speegle D. The geometry of sets of parameters of wave packet frames[J]. Appl Comput Harmon Anal, 2006, 20 (1): 108-125.
[78] Gabardo J P, Han D. The uniqueness of the dual of Weyl-Heisenberg subspace frames[J]. Appl Comput Harmon Anal, 2004, 17(2): 226-240.
[79] Labate D, Wilson E. Connectivity in the set of Gabor frames[J]. Appl Comput Harmon Anal, 2005, 18(1): 123-136.
[80] Benedetto J J, Pfander G E. Frame expansions for Gabor multipliers[J]. Appl Comput Harmon Anal, 2006, 20(1): 26-40.
[81]李松.在(L p (R s ))r(1≤p≤∞)中的向量细分格式[J].中国科学(A缉), 2003, 33(1): 1-11.
[82] Jia R, Jiang Q, Shen Z. Convergence of cascade algorithm associated with nonhomogeneous refinement equations[J]. Proc Amer Math Soc, 2001(2), 129: 415-427.
[83] Micchelli C A, Sauer T. On vector subdivision[J]. Math Z, 1998, 229: 621-674.
[84] Jia R, Riemenschneider S D, Zhou D. Vector subdivision schemes and multiple wavelets[J]. Mathematics of Computation, 1998, 67(224): 1533-1563.
[85]李云章,田雄飞.一类二维不可分非正交小波包.应用数学学报[J]. 2001, 24(3): 418-424.
[86] Chen Q, Cheng Z. A study on compactly supported orthogonal vector-valued wavelets and wavelet packets[J]. Chaos, Solitons & Fractals, 2007, 31(4): 1024-1034.
[87] Chang C S, Jin J. Separation of corona using wavelet packet transform and neural network for detection of partial discharge in gasinsulated substations[J]. IEEE Trans Power Delivery, 2005, 20(2): 1363-1369.
[88] He H, Cheng S. Home network power-line communication signal processing based on wavelet packet analysis[J]. IEEE Trans Power Delivery, 2005, 20(3): 1879-1885.
[89] Chui C K, Quak E. Wavelets on a bounded interval[R]. CAT Report # 265, Texas: Texas A & M University, 1992.
[90] Micchelli C A, Sauer T. Regularity of multiwavelets[J]. Adv Comput Math, 1997, 7(4): 455-545.
[91] Micchelli C A, Xu Y. Reconstruction and decomposition algorithms for biorthogonal multiwavelets[J]. Multidimensional Systems and Signal Process, 1997, 8(1): 31-69.
[92]崔丽鸿.紧支撑的小波、多小波理论及构造[D].西安:西安交通大学, 2004.
[93]关履泰.在[0,1]区间的自由结点样条小波[J].工程数学学报, 1998, 15(3): 1-6..
[94] Baggett L, Medina H, Merrill K. Generalized multiresolution analysis, and aconstruction procedure for all wavelet sets in R d[J]. Fourier Anal Appl, 1999, 5(6): 563-573.
[95] Walter G G. A sampling theorem for wavelet subspaces[J]. IEEE Trans Inform Theory, 1992, 38(2), 881-884.
[96] Bownik M. The structure of shift-invariant subspace of L2 ( R d)[J]. Jour Funct Anal, 2000, 177(2): 282-309.
[97] Baki? D. Semi-orthogonal Parseval frame wavelet and generalized multiresolution analyses[J]. Appl Comput Harmon Anal, 2006, 21(3): 281-304.
[98] Dai X. Diao Y, Gu Q, Han D. Frame wavelet sets in R d[J]. Jour Comput Appl Math, 2003, 155(1): 69-82.
[99] Walter G G. Translation and dilation invariance in orthogonal wavelet[J]. Appl Comput Harmon Anal, 1994, 1(4): 344-349.
[100] Walter G G. Wavelet and Other Orthogonal Systems with Applications[M]. Boca Raton, FL: CRC Press, 1994.
[101] Zhang Z. Supports of Fourier transforms of scaling functions[J]. Appl Comput Harmon Anal, 2007, 22(2): 141-156.
[102] Zhou X, Sun W. On the Sampling Theorem for Wavelet Subspaces[J]. Jour Fourier Anal Appl, 1999, 5(4): 347-354.
[103] Ron A, Shen Z. Frames and stable bases for shift-invariant subspaces of L2 ( R d)[J]. Canad J Math, 1995, 47(5): 1051-1094.
[104] Sun W, Zhou X. Sampling theorems for wavelet subspaces: error estimate and irregular sampling[J]. IEEE Trans Signal Process, 2000, 48(1): 223-226.
[105] Chen W, Itoh S, Shiki J. A sampling theorem for shift-invariant spaces[J]. IEEE Trans. Signal Processing, 1998, 46(10): 2802-2810.
[106] Zhao P, Liu G Z, Zhao C. Frames and Sampling Theorems for Translation-Invariant Subspaces[J]. IEEE Trans Signal Processing, 2004, 11(1): 8-11.
[107]龙瑞麟.高维小波分析[M].北京:世界图书出版公司, 1995.
[108] W Li, Zhang Y Q. A Study on the optimal attributes of transform domain vector quantization for image and video compression[R]. ICC, 1993.
[109] Sun L, Cheng Z X. Construction of a class of compactly supported orthogonal vector-valued wavelets[J]. Chaos, Solitons & Fractals, 2007, 34(2): 253-261.
[110] Shannon C E. Communication in the prensence of noise[J]. Proc Institute of Radio Engineers, 1949, 37 (1): 10–21.
[111] Sun W. Sampling theorems for multivariate shift invariant subspaces[J]. Sampl Theory Signal Image Process, 2005, 4(1): 73-98.
[112] Blanco C, Cabrelli C, Heineken S. Functions in Sampling Spaces[J], Sampl Theory Signal Image Process, 2005, 5(3): 275-265.
[113] Hernández E, Weiss G. A First Course on Wavelets[M]. Boca Raton, FL: CRC Press, 1996.
[114] Fornaiser M, Gori L. Sampling theorems on bounded domains[J]. Journal of Computational and Applied Mathematics, 2008, 221(2): 376-385.
[115] Zhao J, Tao R, Wang Y. Sampling rate conversion for linear canonical transform[J]. Signal Processing, 2008, 88(11): 2825-2832.
[116] Chen Q, Wang Y B, Wang Y. A sampling theorem for non-bandlimited signals using generalized Sinc functions[J]. Computer & Mathematics with Applications, 2008, 56(6): 1650-1661.
[117] Annaby M H, Bustoz J, Ismail M E H. On sampling theory and basic Sturm-Liouville systems[J]. Jounral of Computational and Applied Mathematics, 2007, 206(1): 73-85.
[118] Fenoy M M, Ibarrola P. The optional sampling theorem for submartingales in the sequentially planned context[J]. Statistics & Probability Letters, 2007, 77(8): 826-831.
[119] Benedetto J J, Sumetkijakan S. Tight frames and geometric properties of wavelet sets [J]. Adv Comput Math, 2006, 24(1): 35-56.
[120] Ron A, Shen Z. Weyl-Heisenberg frames and Riesz bases in L2 (R d)[J]. Duke Math J, 1997, 89(2): 237-282.
[121] Baki? D, Krishtal I, Wilson E N. Parseval frame wavelets with E n(2)-dilations[J]. Appl Comput Harmon Anal, 2005, 19(3): 386-431.
[122] Kim H O, Kim R Y, Lim J K. On the spectrum of frame multiresolution analysis[J]. J Math Anal Appl, 2005, 305 (2): 528-545.
[123] Dai X, Larson D. Wandering Vectors for Unitary Systems and Orthogonal Wavelets[J]. Memoirs of AMS, 1998, 136(651): No. 640,.
[124] Dai X, Diao Y, Gu Q. Subspaces with Normalized Tight Frame Wavelets in L2 (R )[J]. Proceedings of AMS, 2002, 130(6):1661-1667.
[125] Dai X, Diao Y, Gu Q, Han D. The existence of subspace wavelet sets[J]. Jounral of Computational and Applied Mathematics, 2003, 155(1): 83-90.
[126] Welland G V (Ed). Beyond Wavelets[M]. San Diego, CA: Academic Press , 2003.
[127] Candès E J, Donoho D L. Ridgelets: a key to higher-dimensional intermittency?[J]. Philos Trans Roy Soc, 1999, 357(1760): 2495-2509.
[128] Coifman R R, Meyer F G. Brushlets: A tool for directional image analysis and image compression[J]. Appl Comput Harmon Anal, 1997, 4(2): 147-187.
[129] Candès E J, Donoho D L. New tight frames of curvelets and optimal representations of objects with C 2 singularities[J]. Comm Pure Appl Math, 2004, 57(2): 219-266.
[130] Xia X G, Zhang Z. On sampling theorem , wavelet and wavelet transforms[J]. IEEE Trans Signal Processing, 1993, 41(12): 3524-3535.
[131] Janssen A J E M. The Zak transform and sampling theorems for wavelet subspaces[J]. IEEE Trans Signal Processing, 1993, 41(12): 3360-3365.
[132] Liu Y. Irregular sampling for spline wavelet subspaces[J]. IEEE Trans Inform Theory, 1996, 42(2): 623-627.
[133] Liu Y, Walter G G. Irregular sampling in wavelet subspaces[J]. Jour Fourier Anal Appl, 1995, 2(2): 181-189.
[134] Ojokovic I, Vaidyanatha P P. Generalized sampling theorems in multiresolution subspaces[J]. IEEE Trans Signal Processing, 1997, 45(3): 583-599.
[135] Chen W, Itoh S, Shiki J. Irregular sampling theorems for wavelet subspace[J]. IEEE Trans Inform Theory, 1998, 44(3): 1131-1142.
[136] Young R. An introduction to nonharmonic Fourier series[M]. New York: Academic Press, 1980.
[137] Walter G G. Wavelet subspaces with oversampling property[J]. Indag Math, 1993, 4(4): 499-507.
[138] Zayed A I, Benedetto J J, Paulo J S G, Eds. Shannon-type wavelets and the convergence of their associated wavelet series[A]. In Modern sampling theory[M], Boston: Brikh?user, 2001: 135-152.
[139] Li W. Vector transform and image coding[J]. IEEE Trans Circuits Syst Video Technol, 1991, 1(4): 297-307.
[140] Li W, Zhang Y Q. A Study on the optimal attributes of transform domain vector quantization for image and video compression[R]. ICC, 1993.
[141] Sun L, Cheng Z X. Construction of a class of compactly supported orthogonal vector-valued wavelets[J]. Chaos, Solitons & Fractals, 2007, 34(2): 253-261.
[142] Zhao P, Liu G, Zhao C. Frames and sampling theorems for translation-invariant subspaces[J]. IEEE Signal Process Lett, 2004, 11(1): 8–11.
[143] Aldroubi A, Grochenig K. Non-uniform sampling in shift-in-variant spaces[J]. SIAM Rev, 2001, 43(4): 585–620.
[144] Sun W, Zhou X. Frames and sampling theorem[J]. Sci China, 1998, 41(2): 606–612.
[145] Zhao P, Zhao C, Casazza P G. Perturbation of Regular Sampling in Shift-Invariant Spaces for Frames[J]. IEEE Trans Inf Theory, 2006 52(10): 4643–4648.
[146] Geronimo J S, Hardin D P, Massopust P R. Fractal functions and wavelet expansions based on several scaling functions[J]. J Approx Theory, 1994, 78(3): 373–401.
[147] Gao X. Notes on sampling theorem for multiwavelet[A]. in Proc IEEE ICSP[C], 2002, vol 1, Aug 26–30, 155–157.
[148] Selesnick I W. Interpolating multiwavelet bases and the sampling theorem[J]. IEEE Trans Signal Processing, 1999, 47(6): 1615–1621.
[149] Zhou D. Interpolatory orthogonal multiwavelets and refinable functions[J]. IEEE Trans Signal Processing, 2002, 50(3): 520–527.
[150] Zhao C, Zhao P. Sampling Theorem and Irregular Sampling Theorem for Multiwavelet Subspaces[J]. IEEE Trans Signal Processing, 2005, 53(2): 705–713.
[151] Benedetto J J, Li S. Multiresolution analysis frames with subspaces[A]. In Proc ICASSP 3[C], Minneapolis MN, 1993: 304-307.