Laguerre超群上的广义小波变换与Radon变换求逆公式
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摘要
M.M.Nessibi和K.Trimèche研究了海森堡群H_n上平方可积径向函数空间，在其基础流形上建立了广义小波变换理论；并利用所引进的广义小波，得到Radon变换在该空间上的一个逆公式．本文针对H_n上平方可积柱径向函数空间，讨论了与之相类似的问题．
在第一章中，我们介绍了与小波变换有关的基本概念与一些常用结论．
在第二章中，我们利用Gelfand变换刻画了海森堡群H_n上平方可积柱径向函数空间的基础流形Laguerre超群X=R_+~n×R上的广义小波，讨论了该超群上的广义小波变换与广义小波包变换．
在第三章中，我们利用Laguerre超群上广义小波变换的方法得到了Radon变换在该空间上的一个求逆公式．
M. M. Nessibi and K. Trimeche studied the theory of generalized wavelet transform on the underlying manifold of square integrable radial functions space on the Heisen-berg group Hn. They constructed an inversion formula of the Radon transform on the underlying manifold by means of generalized wavelet. In this paper, we deal with the analogous problems of square integrable polyradial functions space on Hn.
In chapter one, we present some basic concepts and fundamental results about wavelet transform in common sense.
In chapter two, we first give the condition of generalized wavelets on L2(X) by using the Gelfand transform, where X = R+R denotes the underlying manifold of square integrable polyradial functions on the Heisenberg group Hn. Then, we establish the theory of generalized wavelet transform and generalized wavelet packet transform on it.
In chapter three, as an application of the theory of generalized wavelet transform, we obtain an inversion formula of the Radon transform on X .
在第一章中，我们介绍了与小波变换有关的基本概念与一些常用结论．
在第二章中，我们利用Gelfand变换刻画了海森堡群H_n上平方可积柱径向函数空间的基础流形Laguerre超群X=R_+~n×R上的广义小波，讨论了该超群上的广义小波变换与广义小波包变换．
在第三章中，我们利用Laguerre超群上广义小波变换的方法得到了Radon变换在该空间上的一个求逆公式．
M. M. Nessibi and K. Trimeche studied the theory of generalized wavelet transform on the underlying manifold of square integrable radial functions space on the Heisen-berg group Hn. They constructed an inversion formula of the Radon transform on the underlying manifold by means of generalized wavelet. In this paper, we deal with the analogous problems of square integrable polyradial functions space on Hn.
In chapter one, we present some basic concepts and fundamental results about wavelet transform in common sense.
In chapter two, we first give the condition of generalized wavelets on L2(X) by using the Gelfand transform, where X = R+R denotes the underlying manifold of square integrable polyradial functions on the Heisenberg group Hn. Then, we establish the theory of generalized wavelet transform and generalized wavelet packet transform on it.
In chapter three, as an application of the theory of generalized wavelet transform, we obtain an inversion formula of the Radon transform on X .
引文
[1] D. Gabor, Theory of communication, J. Inst. Elect. Eng. (London), 1946(93), 429-457.
[2] A. Grossmann, J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal., 1984(15), 723-736.
[3] J. X. He, H. P. Liu, Admissible wavelets associtated with the affine automorphism group of the Siegel upper half-plane, J. Math. Anal. Appl., 1997(208), 58-70.
[4] H. P. Liu, L. Z. Peng, Admissible wavelets associtated with the Heisenberg group, Pacific J. Math., 1997(180), 101-123.
[5] 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，1999(第42 卷第1期)，77-82。
[6] G. Mauceri, Zonal multiplies on the Heisenberg group, Pacific J. Math., 1981(95), 143-159.
[7] D. M(?)ller, F. Ricci, E. M. Stein, Marcinkiewicz multiplier and multi-parameter structure on Heisenberg(-type) groups, I. Invent. Math., 1995(119), 199-233.
[8] E. M. Stein, Harmonic analysis: resl-variables methods, orthogonality, and oscillatory integrals, (Princenton), Princenton University press, 1993.
[9] K. Stempak, Mean summability methods for Laguerre series, Trans. Amer. Math. Soc., 1990(332), 671-690.
[10] K. Stempak, An algebra associated with the generalized sublaplacian, Studia Math., 1988(85), 245-256.
[11] M. M. Nessibi, K. Trim(?)che, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl., 1997(208), 337-363.
[12] H. G. Feichtinger, K. H. Gr(?)chenig, Banach spaces related to integrable group representations and their atomic decompositions, J. Funct. Anal., 1989(86), 307-340.
[13] A. Grossmann, J. Morelt, T. Paul, Transforms associated to sequare integrable group representations I, J. Math. Phys., 1985(26), 2473-2479.
[14] K. Mokni, K. Trim(?)che, Wavelet transforms on compact Gelfand pair and its discretization, J. Math. Anal. Appl., 1999(238), 234-258.
[15] J. Radon, (?)ber die Bestimmung von Funktionen durch ihre integralwerte l(?)ngs gewisser Mannigfaltigkeiten, Berichte Sachsische Akademie der Wissenschaften, Leipzig, Math. Phys. K. I., 1917(69), 262-267.
[16] D. Ludwig, The Radon transform on Euclidean space, Comm. Pure. Apple. Math., 1966(19), 49-81.
[17] S. Helgason, The Radon transform, (Boston), Birkh(?)user, 1980.
[18] S. R. Deans, The Radon transform and some of its application, (New York), Wiley, 1983.
[19] 李远钦，刘雯林，n维广义Radon变换，地球物理学进展，1997(第12卷第4期)，59-66．
[20] R. S. Strichartz, L~p harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal., 1991(96), 350-406.
[21] G. T. Herman, Image reconstruction form projections, the fundamentals of computerized tomography, Academic Press, 1980.
[22] 刘秀君，郑克旺，Radon变换的一个反演公式，河北轻化工学院学报，1997(第18卷第3期)，49-51．
[23] H. Stark, J. W. Woods, I. P. Paul, R. HIngorani, An investigation of computerized tomography by direct Fourier inversion and optimun interpolation, IEEE Trans. Biomed. Engin., 1981(BME-28), 496-505.
[24] F. Natterer, The mathematics of computerized tomography, (New York), Wiley, 1986.
[25] 王金平，平面Radon变换的反演公式，CT理论与应用研究，2000(第24卷第4期)，8-11．
[26] 陈东方，基于小波变换的Radon变换反演，安徽大学学报(自然科学版)，2000(第24卷第4期)，24-33．
[2] A. Grossmann, J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal., 1984(15), 723-736.
[3] J. X. He, H. P. Liu, Admissible wavelets associtated with the affine automorphism group of the Siegel upper half-plane, J. Math. Anal. Appl., 1997(208), 58-70.
[4] H. P. Liu, L. Z. Peng, Admissible wavelets associtated with the Heisenberg group, Pacific J. Math., 1997(180), 101-123.
[5] 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，1999(第42 卷第1期)，77-82。
[6] G. Mauceri, Zonal multiplies on the Heisenberg group, Pacific J. Math., 1981(95), 143-159.
[7] D. M(?)ller, F. Ricci, E. M. Stein, Marcinkiewicz multiplier and multi-parameter structure on Heisenberg(-type) groups, I. Invent. Math., 1995(119), 199-233.
[8] E. M. Stein, Harmonic analysis: resl-variables methods, orthogonality, and oscillatory integrals, (Princenton), Princenton University press, 1993.
[9] K. Stempak, Mean summability methods for Laguerre series, Trans. Amer. Math. Soc., 1990(332), 671-690.
[10] K. Stempak, An algebra associated with the generalized sublaplacian, Studia Math., 1988(85), 245-256.
[11] M. M. Nessibi, K. Trim(?)che, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl., 1997(208), 337-363.
[12] H. G. Feichtinger, K. H. Gr(?)chenig, Banach spaces related to integrable group representations and their atomic decompositions, J. Funct. Anal., 1989(86), 307-340.
[13] A. Grossmann, J. Morelt, T. Paul, Transforms associated to sequare integrable group representations I, J. Math. Phys., 1985(26), 2473-2479.
[14] K. Mokni, K. Trim(?)che, Wavelet transforms on compact Gelfand pair and its discretization, J. Math. Anal. Appl., 1999(238), 234-258.
[15] J. Radon, (?)ber die Bestimmung von Funktionen durch ihre integralwerte l(?)ngs gewisser Mannigfaltigkeiten, Berichte Sachsische Akademie der Wissenschaften, Leipzig, Math. Phys. K. I., 1917(69), 262-267.
[16] D. Ludwig, The Radon transform on Euclidean space, Comm. Pure. Apple. Math., 1966(19), 49-81.
[17] S. Helgason, The Radon transform, (Boston), Birkh(?)user, 1980.
[18] S. R. Deans, The Radon transform and some of its application, (New York), Wiley, 1983.
[19] 李远钦，刘雯林，n维广义Radon变换，地球物理学进展，1997(第12卷第4期)，59-66．
[20] R. S. Strichartz, L~p harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal., 1991(96), 350-406.
[21] G. T. Herman, Image reconstruction form projections, the fundamentals of computerized tomography, Academic Press, 1980.
[22] 刘秀君，郑克旺，Radon变换的一个反演公式，河北轻化工学院学报，1997(第18卷第3期)，49-51．
[23] H. Stark, J. W. Woods, I. P. Paul, R. HIngorani, An investigation of computerized tomography by direct Fourier inversion and optimun interpolation, IEEE Trans. Biomed. Engin., 1981(BME-28), 496-505.
[24] F. Natterer, The mathematics of computerized tomography, (New York), Wiley, 1986.
[25] 王金平，平面Radon变换的反演公式，CT理论与应用研究，2000(第24卷第4期)，8-11．
[26] 陈东方，基于小波变换的Radon变换反演，安徽大学学报(自然科学版)，2000(第24卷第4期)，24-33．