乘积海森堡群及拉盖尔超群上的多尺度分析
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摘要
本文定义了乘积海森堡群及拉盖尔超群上的多尺度分析,并且研究了乘积海森堡群及拉盖尔超群上的L~2空间的Haar小波基的性质.另外本文还给出了一种构造乘积海森堡群上的L~2空间的Haar小波基的方法.本文的第一章主要介绍了海森堡群及拉盖尔超群的来源,历史背景和现在已有的结果,并且为了证明结论的方便我们介绍了一些要用的预备知识.第二章首先定义了乘积海森堡群上的多尺度分析然后研究了乘积海森堡群上的L~2空间的Haar小波基的性质及怎样构造Haar小波基.第三章重点研究拉盖尔超群上的多尺度分析及其上的L~2空间的Haar小波基的性质.
In this paper the properties of multiresolution analysis on the product of Heisenberg group and Laguerre hypergroup are investigated.Moreover,the properties of Haar wavelet bases for L~2 space on the product of Heisenberg group and Laguerre hypergroup are investigated.Orthonormal Haar wavelet bases for L~2 space on the product of Heisenberg group are also constructed.In chapter 1,the author introduces the historical background and the recent development of problems to be studied about Heisenberg group and Laguerre hypergroup in details and some preparations are given here.In chapter 2,firstly,the author defines the properties of multiresolution analysis on the product of Heisenberg group.And then orthonormal Haar wavelet bases for L~2 space on the product of Heisenberg group are constructed.In chapter 3,the multiresolution analysis on the Laguerre hypergroup is defined.Moreover the properties of Haar wavelet bases for L~2 space on the Laguerre hypergroup are investigated.
In this paper the properties of multiresolution analysis on the product of Heisenberg group and Laguerre hypergroup are investigated.Moreover,the properties of Haar wavelet bases for L~2 space on the product of Heisenberg group and Laguerre hypergroup are investigated.Orthonormal Haar wavelet bases for L~2 space on the product of Heisenberg group are also constructed.In chapter 1,the author introduces the historical background and the recent development of problems to be studied about Heisenberg group and Laguerre hypergroup in details and some preparations are given here.In chapter 2,firstly,the author defines the properties of multiresolution analysis on the product of Heisenberg group.And then orthonormal Haar wavelet bases for L~2 space on the product of Heisenberg group are constructed.In chapter 3,the multiresolution analysis on the Laguerre hypergroup is defined.Moreover the properties of Haar wavelet bases for L~2 space on the Laguerre hypergroup are investigated.
引文
[1]I.Daubechies,Ten Lectures on Wavelets(SIAM,1992).
[2]D.Geller,Fourier analysis on the Heisenberg group,J.Funet Anal.36(1980)205-254.
[3]K.Gr(o|¨)chenig and W.R.Madych,Multiresolution analysis,Haar bases and selfsimilar filings of R~n,IEEE.Transactions on Information Theory.38(1992) 556-568.
[4]J.X.He,An Inversion formula of the Radon Transform on the Heisenberg group,Canad.Math.Bull.47(2004) 389-397.
[5]J.X.He,Continuous Multiscale Analysis on the Heisenberg Group,Bull Korean.Math.Soc.38(2001) 517-526.
[6]J.X.He and H.P.Liu,Admissible Wavelets Associated with the Affine Automorphism Group of the Siegel Upper Half-Plane,J.Math.Anal Appl.208(1997)58-70.
[7]W.M.Lawton,Infinite convolution products and refinable distributions on Lie groups,Trans.Amer.Math.Soc.352(2000) 2913-2936.
[8]H.P.Liu and L.Z.Peng,Admissible wavelets associated with the Heisenberg Group,J.Pacific.Math.180(1997) 101-123.
[9]H.P.Liu,Y.Liu and H.H.Wang,Multiresolution analysis,self-similar tilings and Haar wavelets on the Heisenberg group,Acta Math.Sci.29B(2009) to appear.
[10]S.Mallat,Multiresolution approximations and wavelet orthonormal bases of L~2(R),Trans.Amer.Math.Soc.315(1989) 69-88.
[11]Sarita Azal,R.Narasimha and S.K.Sett,Multiresolution analysis for separating closely spaced frequencies with an application to Indian monsoon rainfull data,Inter.J.Wavelets,Multiresolution and lnformation Processing.5(2007) 735-752.
[12]R.S.Strichartz,Wavelet and self-affine tilings,Constr.Approx.9(1993) 327-346.
[13]S.Thangavelu,Restriction theorems for the Heisenberg group,J.reine angew.Math.414(1991) 51-65.
[14]W.P.Thurston,Groups,Tilings,and Finite State Automata,in Lecture Notes,Summer Meetings Amer.Math.Soc.1989.
[15]P.Z.Xie and J.X.He,Multiresolution analysis and Haar wavelets on the product of Heisenberg group,Int.J.Wavelets Mult.Inf.Proc.,to appear,2009.
[16]M.Assal and H.Ben Abdallah,Generalized Besov type spaces on the Laguerre hypergroup,J.Anal Math.Blaise Pascal,vol.12,pp.117-145,2005.
[17]M.M.Nessibi and K.Trimeche,Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets,J.Math.Anal Appl,vol.208,pp.337-363,1997.
[18]何建勋,彭立中,群上的带状小波及函数空间的正交分解,数学学报,Vol.42(1)(1999)77-82.
[19]G.Mauceri,Zonal Multipliers on the Heisenberg Group,Pacific J.Math,95(1981),143-159.
[20]K.Stempak,Mean summability methods for Laguerre series,Trans.Amer.Math.Soc.,322(1990),671-690.
[21]D.M(u|¨)ller and E.M.Stein,On Spectral Multiplier for the Heisenberg and Related Groups,J.Math.Pures.Appl,73(1994),413-440.
[22]E.M.Stein,Harmonic Analysis:Real-Variables Methods,Orthogonality,and Oscillatory Integrals,(Princeton University Press,Princeton,1993)
[23]K.Stempak,An algebra associated with the generalized subaplacian,Studia Math.,88(1988),245-256.
[2]D.Geller,Fourier analysis on the Heisenberg group,J.Funet Anal.36(1980)205-254.
[3]K.Gr(o|¨)chenig and W.R.Madych,Multiresolution analysis,Haar bases and selfsimilar filings of R~n,IEEE.Transactions on Information Theory.38(1992) 556-568.
[4]J.X.He,An Inversion formula of the Radon Transform on the Heisenberg group,Canad.Math.Bull.47(2004) 389-397.
[5]J.X.He,Continuous Multiscale Analysis on the Heisenberg Group,Bull Korean.Math.Soc.38(2001) 517-526.
[6]J.X.He and H.P.Liu,Admissible Wavelets Associated with the Affine Automorphism Group of the Siegel Upper Half-Plane,J.Math.Anal Appl.208(1997)58-70.
[7]W.M.Lawton,Infinite convolution products and refinable distributions on Lie groups,Trans.Amer.Math.Soc.352(2000) 2913-2936.
[8]H.P.Liu and L.Z.Peng,Admissible wavelets associated with the Heisenberg Group,J.Pacific.Math.180(1997) 101-123.
[9]H.P.Liu,Y.Liu and H.H.Wang,Multiresolution analysis,self-similar tilings and Haar wavelets on the Heisenberg group,Acta Math.Sci.29B(2009) to appear.
[10]S.Mallat,Multiresolution approximations and wavelet orthonormal bases of L~2(R),Trans.Amer.Math.Soc.315(1989) 69-88.
[11]Sarita Azal,R.Narasimha and S.K.Sett,Multiresolution analysis for separating closely spaced frequencies with an application to Indian monsoon rainfull data,Inter.J.Wavelets,Multiresolution and lnformation Processing.5(2007) 735-752.
[12]R.S.Strichartz,Wavelet and self-affine tilings,Constr.Approx.9(1993) 327-346.
[13]S.Thangavelu,Restriction theorems for the Heisenberg group,J.reine angew.Math.414(1991) 51-65.
[14]W.P.Thurston,Groups,Tilings,and Finite State Automata,in Lecture Notes,Summer Meetings Amer.Math.Soc.1989.
[15]P.Z.Xie and J.X.He,Multiresolution analysis and Haar wavelets on the product of Heisenberg group,Int.J.Wavelets Mult.Inf.Proc.,to appear,2009.
[16]M.Assal and H.Ben Abdallah,Generalized Besov type spaces on the Laguerre hypergroup,J.Anal Math.Blaise Pascal,vol.12,pp.117-145,2005.
[17]M.M.Nessibi and K.Trimeche,Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets,J.Math.Anal Appl,vol.208,pp.337-363,1997.
[18]何建勋,彭立中,群上的带状小波及函数空间的正交分解,数学学报,Vol.42(1)(1999)77-82.
[19]G.Mauceri,Zonal Multipliers on the Heisenberg Group,Pacific J.Math,95(1981),143-159.
[20]K.Stempak,Mean summability methods for Laguerre series,Trans.Amer.Math.Soc.,322(1990),671-690.
[21]D.M(u|¨)ller and E.M.Stein,On Spectral Multiplier for the Heisenberg and Related Groups,J.Math.Pures.Appl,73(1994),413-440.
[22]E.M.Stein,Harmonic Analysis:Real-Variables Methods,Orthogonality,and Oscillatory Integrals,(Princeton University Press,Princeton,1993)
[23]K.Stempak,An algebra associated with the generalized subaplacian,Studia Math.,88(1988),245-256.