乘积海森堡群上的逆Radon变换以及魏尔变换
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摘要
令H1是3-维的海森堡群,称Hn1 = H1×H1×...×H1为乘积海森堡群.本文讨论乘积海森堡群上的广义小波变换,利用小波变换的反演公式得到了Radon变换的逆公式.其次,结合小波变换研究了魏尔变换的有界性.全文共分为三章:
第一章,介绍本文的学术背景.
第二章,研究了乘积海森堡群上与Schro¨dinger表示相应的傅里叶变换以及小波变换,并利用小波变换的反演公式得到Radon变换的逆公式.
第三章,研究了乘积海森堡群上与Bargmann-Fock表示相应的傅里叶变换以及小波变换的性质,并讨论与小波变换联系的魏尔变换的有界性.
Let H1 be the real 3-diemensional Heisenberg group, then we call Hn1 = H1×H1×...×H1 the product Heisenberg group. In this article, we first discuss thegeneralized wavelet transform of the product Heisenberg group, and use the inversewavelet transform to get an inversion formula of Radon transform. In addition, westudy the boundedness of Weyl transform associated with the wavelet transform. Itconsists of 3 chapters:
In Chapter 1, the academic background is introduced.
In Chapter 2, we study the Fourier transform defined by Schro¨dinger representa-tions and construct the wavelet transform on the product Heisenberg group. Moreover,we obtain an inversion formula of Radon transform by using the inverse wavelet trans-form.
In Chapter 3, we study the Fourier transform related to the Bargmann-Fock rep-resentations and the wavelet transform on the product Heisenberg group. Furthermore,we investigate the boundedness of Weyl transform associated with the wavelet trans-form.
第一章,介绍本文的学术背景.
第二章,研究了乘积海森堡群上与Schro¨dinger表示相应的傅里叶变换以及小波变换,并利用小波变换的反演公式得到Radon变换的逆公式.
第三章,研究了乘积海森堡群上与Bargmann-Fock表示相应的傅里叶变换以及小波变换的性质,并讨论与小波变换联系的魏尔变换的有界性.
Let H1 be the real 3-diemensional Heisenberg group, then we call Hn1 = H1×H1×...×H1 the product Heisenberg group. In this article, we first discuss thegeneralized wavelet transform of the product Heisenberg group, and use the inversewavelet transform to get an inversion formula of Radon transform. In addition, westudy the boundedness of Weyl transform associated with the wavelet transform. Itconsists of 3 chapters:
In Chapter 1, the academic background is introduced.
In Chapter 2, we study the Fourier transform defined by Schro¨dinger representa-tions and construct the wavelet transform on the product Heisenberg group. Moreover,we obtain an inversion formula of Radon transform by using the inverse wavelet trans-form.
In Chapter 3, we study the Fourier transform related to the Bargmann-Fock rep-resentations and the wavelet transform on the product Heisenberg group. Furthermore,we investigate the boundedness of Weyl transform associated with the wavelet trans-form.
引文
[1] Linglan Bai and Jianxun He, Generalized wavelets and inversion of the Radontransform on the Laguerre hypergroup, Approx.Theory and its Appl. 18(4)(2002),55-69.
[2] P. Boggiatto, E. Buzano, L.Rodino, Global Ellipticity and Spectral Theory, Math-ematical Research, 92(1996), Akademie-Verlag.
[3] P. Boggiatto, L.Rodino, Quantization and pseudo-differential operators, CuboMathematica Educationsl , 1(2003).
[4] I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[5] H. G. Feichtinger and K.H.Gro¨cheng, Banach Space Related to Itegrable GroupRepresentations and Their Atomic Decompositions, J. Funct. Anal. 86(1989), 307-340.
[6] D. Geller, Fourier analysis on the Heisenberg group, J. Funct Anal. 36 (1980)205-254.
[7] D. Geller, Fourier analysis on the Heisenberg group. l. Schwartz space, Journalof Functional Analysis, 36(2)(1980), 255-271.
[8] K.H. Gro¨cheng, Foundation of time-frequency analysis, Birkha¨user, Boston,2001.
[9] S. Helgason, The Radon Transform, 2nd edn. Birkha¨user, Boston, Basel, Berlin,1999.
[10] M. Holschneider, Inverse Radon Transforms Through Inverse Wavelet Trans-forms, Inverse Problems 7(1991), 853-861.
[11] Qingtang Jiang, Rotation invariant ambiguity function, Proc. Amer. Math. Soc126(1996), 561- 567.
[12] Jianxun He, An Inversion formula of the Radon Transform on the Heisenberggroup, Canad. Math. Bull. 47 (2004) 389-397.
[13] Jianxun He, A characterization of inverse Radon transform on the Laguerre hy-pergroup, J. Math. Anal. Appl. 318(2006), 387-395.
[14] Jianxun He and Heping Liu, Admissible Wavelets Associated with the AffineAutomorphism Group of the Siegel Upper Half-Plane, J. Math. Anal. Appl.208(1997), 58-70.
[15] Jianxun He and Heping Liu, Inversion of the Radon Transform Associated withthe Classical Domain of Type One, Inter. J. Math. 16(2005), 875-887.
[16] Jianxun He and Liqun Wen, An inversion formula of Radon transform on theproduct Heisenberg group, accepted by Rocky Mountain Journal of Mathematics.
[17]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,42(1) (1999), 77-82.
[18] W. M. Lawton, Infinite convolution products and refinable distributions on Liegroups, Trans. Amer. Math. Soc. 352(2000), 2913-2936.
[19] Heping Liu and Lizhong Peng, Admissible wavelets associated with the Heisen-berg group, Pac. J. Math. 180(1997), 101-121.
[20] D. Ludwig, The Radon transform on Euclidean space, Comm.Pure Appl.Math.23(1966), 49-81.
[21] G. Mauceri, Zonal Multipliers on the Heisenberg Group, Pacific J. Math,95(1981), 143-159.
[22] Ruiqin Ma and Lizhong Peng, Weyl transform associated with wavelets,Progress in Natural Science, 13(7)(2003), 497-500.
[23] D. Mu¨ller and E. M. Stein,On Spectral Multiplier for the Heisenberg and RelatedGroups, J. Math. Pures. Appl, 73(1994), 413-440.
[24] M. M. Nessibi and K.Trine`che, Inversion of the Radon Transforms on theLaguerre Hypergroup by Using Generalized Wavelets, J. Math. Anal. Appl.208(1997), 337-363.
[25] Lizhong Peng and Ruiqin Ma, Wavelets associated with Hankel transform andtheir Weyl transforms, Science in China Ser. A Mathematics, 47(2004), 393-400.
[26] Lizhong Peng and Jiman Zhao, Weyl transforms associated with the Heisenberggroup, Bull. Sci. Math. , 132(1)(2008), 78–86.
[27] J. C. T.Pool, Mathematical aspect of the Weyl correspondence, J. Math. Phys,7(1966), 66- 76.
[28] M. Reed, B. Simon, Methods of Modern Mathematical Physics I: FunctionalAnalysis, Academic Press, New York, 1972.
[29] B. Rubin, The Caldero`n reproducing formula, windowed X-ray transforms andRadon transforms in Lp-spaces, J. Fourier Anal. Appl. 4(1998), 175-197.
[30] B. Rubin, Fractional Calculus and wavelet transforms in integral geometry,Fract. Calculus Appl. Anal. 1(1998), 193-219.
[31] B. Rubin, Convolution-backprojection method for the k-plane transform, andCaldero′n’s identity for ridgelet transforms, Appl. Comput. Harmon. Anal.16(2004), 231-242.
[32] B. Simon, The Weyl transform and Lp functions on phase space, Proceedings ofthe American Mathematical Soceity, 116(1992), 1045-1047.
[33] E. M. Stein, Harmonic Analysis. Real-Variable Methods, Orthogonality, and Os-cillatory Integrals, Princeton Univ. Press, 1993.
[34] K. Stempak, Mean summability methods for Laguerre series, Trans. Amer. Math.Soc., 322(1990),671-690.
[35] R. S. Strichartz, Lp harmonic analysis and Radon transform on the Heisenberggroup, J. Funct. Anal. 96(1991), 350-406.
[36] S. Thangavelu, Restriction theorems for the Heisenberg group, J. reine angew.Math. 414(1991) 51-65.
[37] J. Toft, Continuity properties for modulation spaces with applications to pseu-dodifferential calculus, I, Journal of Functional Analysis, 207(2)(2004), 399-429.
[38] J. Toft, Continuity properties for modulation spaces with applications to pseu-dodifferential calculus, II, Annals of Global Analysis and Geomotry , 26( 2004),73-106.
[39] H. Weyl, The theory of groups and quantum mechanics, Dover, New York, 1931.
[40] M.W. Wong, Weyl transform, Sringer-Verlag, New York, 1998.
[41] Jiman Zhao and Lizhong Peng, Wavelet and Weyl transform associated with thespherical mean operator, Integral Equations and Operator Theory, 50(2004), 279-290.
[42] Zhenqiu Zhang, A restriction for the Product Heisenberg group, Chinese ScienceBulletin, 40(19)(1995), 1589-1591.
[2] P. Boggiatto, E. Buzano, L.Rodino, Global Ellipticity and Spectral Theory, Math-ematical Research, 92(1996), Akademie-Verlag.
[3] P. Boggiatto, L.Rodino, Quantization and pseudo-differential operators, CuboMathematica Educationsl , 1(2003).
[4] I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[5] H. G. Feichtinger and K.H.Gro¨cheng, Banach Space Related to Itegrable GroupRepresentations and Their Atomic Decompositions, J. Funct. Anal. 86(1989), 307-340.
[6] D. Geller, Fourier analysis on the Heisenberg group, J. Funct Anal. 36 (1980)205-254.
[7] D. Geller, Fourier analysis on the Heisenberg group. l. Schwartz space, Journalof Functional Analysis, 36(2)(1980), 255-271.
[8] K.H. Gro¨cheng, Foundation of time-frequency analysis, Birkha¨user, Boston,2001.
[9] S. Helgason, The Radon Transform, 2nd edn. Birkha¨user, Boston, Basel, Berlin,1999.
[10] M. Holschneider, Inverse Radon Transforms Through Inverse Wavelet Trans-forms, Inverse Problems 7(1991), 853-861.
[11] Qingtang Jiang, Rotation invariant ambiguity function, Proc. Amer. Math. Soc126(1996), 561- 567.
[12] Jianxun He, An Inversion formula of the Radon Transform on the Heisenberggroup, Canad. Math. Bull. 47 (2004) 389-397.
[13] Jianxun He, A characterization of inverse Radon transform on the Laguerre hy-pergroup, J. Math. Anal. Appl. 318(2006), 387-395.
[14] Jianxun He and Heping Liu, Admissible Wavelets Associated with the AffineAutomorphism Group of the Siegel Upper Half-Plane, J. Math. Anal. Appl.208(1997), 58-70.
[15] Jianxun He and Heping Liu, Inversion of the Radon Transform Associated withthe Classical Domain of Type One, Inter. J. Math. 16(2005), 875-887.
[16] Jianxun He and Liqun Wen, An inversion formula of Radon transform on theproduct Heisenberg group, accepted by Rocky Mountain Journal of Mathematics.
[17]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,42(1) (1999), 77-82.
[18] W. M. Lawton, Infinite convolution products and refinable distributions on Liegroups, Trans. Amer. Math. Soc. 352(2000), 2913-2936.
[19] Heping Liu and Lizhong Peng, Admissible wavelets associated with the Heisen-berg group, Pac. J. Math. 180(1997), 101-121.
[20] D. Ludwig, The Radon transform on Euclidean space, Comm.Pure Appl.Math.23(1966), 49-81.
[21] G. Mauceri, Zonal Multipliers on the Heisenberg Group, Pacific J. Math,95(1981), 143-159.
[22] Ruiqin Ma and Lizhong Peng, Weyl transform associated with wavelets,Progress in Natural Science, 13(7)(2003), 497-500.
[23] D. Mu¨ller and E. M. Stein,On Spectral Multiplier for the Heisenberg and RelatedGroups, J. Math. Pures. Appl, 73(1994), 413-440.
[24] M. M. Nessibi and K.Trine`che, Inversion of the Radon Transforms on theLaguerre Hypergroup by Using Generalized Wavelets, J. Math. Anal. Appl.208(1997), 337-363.
[25] Lizhong Peng and Ruiqin Ma, Wavelets associated with Hankel transform andtheir Weyl transforms, Science in China Ser. A Mathematics, 47(2004), 393-400.
[26] Lizhong Peng and Jiman Zhao, Weyl transforms associated with the Heisenberggroup, Bull. Sci. Math. , 132(1)(2008), 78–86.
[27] J. C. T.Pool, Mathematical aspect of the Weyl correspondence, J. Math. Phys,7(1966), 66- 76.
[28] M. Reed, B. Simon, Methods of Modern Mathematical Physics I: FunctionalAnalysis, Academic Press, New York, 1972.
[29] B. Rubin, The Caldero`n reproducing formula, windowed X-ray transforms andRadon transforms in Lp-spaces, J. Fourier Anal. Appl. 4(1998), 175-197.
[30] B. Rubin, Fractional Calculus and wavelet transforms in integral geometry,Fract. Calculus Appl. Anal. 1(1998), 193-219.
[31] B. Rubin, Convolution-backprojection method for the k-plane transform, andCaldero′n’s identity for ridgelet transforms, Appl. Comput. Harmon. Anal.16(2004), 231-242.
[32] B. Simon, The Weyl transform and Lp functions on phase space, Proceedings ofthe American Mathematical Soceity, 116(1992), 1045-1047.
[33] E. M. Stein, Harmonic Analysis. Real-Variable Methods, Orthogonality, and Os-cillatory Integrals, Princeton Univ. Press, 1993.
[34] K. Stempak, Mean summability methods for Laguerre series, Trans. Amer. Math.Soc., 322(1990),671-690.
[35] R. S. Strichartz, Lp harmonic analysis and Radon transform on the Heisenberggroup, J. Funct. Anal. 96(1991), 350-406.
[36] S. Thangavelu, Restriction theorems for the Heisenberg group, J. reine angew.Math. 414(1991) 51-65.
[37] J. Toft, Continuity properties for modulation spaces with applications to pseu-dodifferential calculus, I, Journal of Functional Analysis, 207(2)(2004), 399-429.
[38] J. Toft, Continuity properties for modulation spaces with applications to pseu-dodifferential calculus, II, Annals of Global Analysis and Geomotry , 26( 2004),73-106.
[39] H. Weyl, The theory of groups and quantum mechanics, Dover, New York, 1931.
[40] M.W. Wong, Weyl transform, Sringer-Verlag, New York, 1998.
[41] Jiman Zhao and Lizhong Peng, Wavelet and Weyl transform associated with thespherical mean operator, Integral Equations and Operator Theory, 50(2004), 279-290.
[42] Zhenqiu Zhang, A restriction for the Product Heisenberg group, Chinese ScienceBulletin, 40(19)(1995), 1589-1591.