乘积Laguerre超群上的广义小波变换及Radon逆变换
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摘要
令H_1是3-维的海森堡群, H1上径向函数空间的基础流形记为[0,+∞)×R,称为Laguerre超群([25]).自然地, K~n = [0,+∞)~n×R~n称为乘积Laguerre超群.本文首先建立了K~n = [0,+∞)~n×R~n上的平移算子和L2(K~n,dμ)上的Plancherel公式.其次,讨论K~n = [0,+∞)~n×R~n上的广义小波变换和Radon变换理论.然后,我们构造S(K~n)(施瓦茨空间)的一个特征子空间SR(K~n),指出Radon变换在SR(K~n)上是一一映射,并给出与SR(K~n)等价的S(K~n)的另一个特征子空间S?,2(K~n).最后,在弱意义下,利用广义小波逆变换得到K~n = [0,+∞)~n×R~n上Radon变换的逆公式.类似地,此结果在海森堡群上成立.
Let H1 be the 3-dimensional Heisenberg group. The fundamental manifold ofthe radial function space for H1 can be denoted by [0, +∞)×R, which is just theLaguerre hypergroup(see[25]). Naturally, K~n = [0, +∞)~n×R~n is product Laguerrehypergroup. In this paper, we construct a generalized translation operator on K~n =[0,+∞)~n×R~n, and establish the Plancherel formula on L2(K~n,dμ). Then we discussthe continuous wavelets transform and Radon transform on K~n, and we characterizea subspace SR(K~n) of S (K~n)(Schwartz space), on which Radon transform is abijection. Also, we give another characterized subspace S?,2(K~n) which is equivalentto SR(K~n). Using the inverse wavelet transform, we obtain an inversion formulaof Radon transform on K~n in the weak sense. Analogously, the same case can beextended to the Heisenberg group.
Let H1 be the 3-dimensional Heisenberg group. The fundamental manifold ofthe radial function space for H1 can be denoted by [0, +∞)×R, which is just theLaguerre hypergroup(see[25]). Naturally, K~n = [0, +∞)~n×R~n is product Laguerrehypergroup. In this paper, we construct a generalized translation operator on K~n =[0,+∞)~n×R~n, and establish the Plancherel formula on L2(K~n,dμ). Then we discussthe continuous wavelets transform and Radon transform on K~n, and we characterizea subspace SR(K~n) of S (K~n)(Schwartz space), on which Radon transform is abijection. Also, we give another characterized subspace S?,2(K~n) which is equivalentto SR(K~n). Using the inverse wavelet transform, we obtain an inversion formulaof Radon transform on K~n in the weak sense. Analogously, the same case can beextended to the Heisenberg group.
引文
[1] Bai, L. and He, J., Generalized wavelets and inversion of the Radon transformon the Laguerre hypergroup, Approx. Theory and its Appl. Vol. 18(4)(2002),55-69.
[2] Bloom, W. R. and Heyer, H., Harmonic analysis of probability measures onhypergroup,in ”de Gruyter Studies in Mathematics,”, Vol. 20, de Gruyter,Berlin/New York, (1994).
[3] Deans, S. R., The Radon Transform and some of its application, (New York),Wiley, (1983).
[4] Erdelyi, A., Magnus, W., Oberhettionger, F. and Tricomi, F. G., Higher Tran-scendental functions, Vol. 2, McGraw-Hill New York, (1953).
[5] Faraut, J. and Harzallah, K., Deux cours d’Analyse Harmonique, Bikha¨user,Boston, (1984).
[6] Felix, R., Radon-Transform auf nilpotenten Lie-Gruppen. Invent. Math.,112(1993), 413-443.
[7] Geller, D. and Stein, E. M., Singular convolution operation on the Heisenberggroup, Bull. Amer. Math. Soc., 6(1982), 99-103.
[8] Gonzalez, F. B., Radon transform on grassmann manifolds. J. Funct. Anal.,71(1987), 339-362.
[9] Grossmann, A. and Morlet, J., Decomposition of Hardy functions into squareintegrable wavelets of constant shape, SIAM J. Math. Anal., 15(1984), 723-736.
[10] He, J., A characterization of inverse Radon transform on the Laguerre hyper-group, J. Math. Anal. Appl., 318(2006), 387-395.
[11] He, J., An Inversion Formula of the Radon Transform on the Heisenberg Group,Ganad. Math. Bull. Vol. 47(3)(2004), 389-397.
[12] He, J. and Liu,H., Inversion of the Radon Transform Associated with the Classi-cal Domain of Type One, Int. J. Math., 16(2005), 875-887.
[13] He, J., and Liu, H., Admissible Wavelets and Inversion Radon Transform Asso-ciated With the Affine Homogeneous Siegal Domains of Type II, Comm. Anal.Geom. Vol. 15(1), (2007), 1-28.
[14] He, J. and Liu, H., Admissible wavelets associtated with the affine automorphismgroup of the Siegel upper half-plane, J. Math. Anal. Appl., 208(1997), 58-90.
[15] 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.
[16] Helgason, S., The Radon Transform, 2nd edn. (Birkha¨user, Boston, Basel,Berlin, 1999).
[17] Holschneider, M., Inverse Radon transform through inverse wavelet transform,Inverse Problems, 7(1991), 853-861.
[18] Jewett, R. I., Space with an abstract convolution of measures, Adv. in Math.,18(1975), 1-101.
[19] Koorwinder, T. H., The addition formula for Laguerre Polynomials, SLAM J.Math. Anal., 8(1977), 535-540.
[20] Lin Chincheng, Lp Mulitipliers and Their H1 ? L1 Estimates on the HeisenbergGroup, Revista. Mat. Ibero., 11(1995), 269-308.
[21] Liu, H. and Peng, L., Admissible wavelets associtated with the Heisenberggroup, Pacific J. Math., 180(1997), 101-123.
[22] Ludwig, D., The Radon transform on Euclidean space, Comm. Pure Appl. Math.23(1966), 49-81.
[23] Mauceri, G., Zonal Multipliers on the Heisenberg Group, Pacific J. Math.95(1981), 143-159.
[24] Mu¨ller, D. and Stein, E. M., On Spectral Multiplier for the Heisenberg and Re-lated Groups, J. Math. Pures. Appl, 73(1994), 413-440.
[25] Nessibi, M. M. and Trime`che, K., Inversion of the Radon transform on theLaguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl.,208(1997), 337-363.
[26] Nessibi, M. M., Sifi, M. and Trime`che, K., Continuous wavelet transform andcontinuous multiscale analysis on Laguerre hypergroup, C. R. Math. Rep. Acad.Sci. Canada, 17(1995), 73-78.
[27] Rubin, B., The Caldero′n formula,windowed x-ray transform on the Laguerrehypergroup in Lp-space, J. Fourier Anal. Appl., 4(1998), 175-197.
[28] Rubin, B., Fractional calculus and wavelet transforms in integral geometry,Fractional Calculus and Applied Analysis 1(1998), 193-219.
[29] Solmon, D. C., Asymptotic formulas for the dual Radon transform and applica-tions, Math. Z., 195(1987), 321-343.
[30] Stein, E. M., Harmonic Analysis: Real-Variables Methods, Orthogonality, andOscillatory Integrals, Puinceton University Press, Princeton, (1993).
[31] Stempak, K., Mean summability methods for Laguerre series, Trans. Amer.Math. Soc., 322(1990), 671-690.
[32] Stempak, K., Almost everywhere summability of Laguerre series, Studia Math.,100(1991), 129-147.
[33] Stempak, K., An algebra associated with the generalized sublaplacian, StudiaMath., 88(1988), 245-256.
[34] Strichartz, R. S., Lp Harmonic aralysis and Radon transforms on the Heisenberggroup, J. Funct. Anal., 96(1991), 350-406.
[35] Thangavelu, S. Restriction Theorems for the Heisenberg Group, J. Reine Angew.Math., 414(1991), 51-65.
[36] 李远钦, 刘雯林, n维广义Radon变换, 地球物理学进展, Vol. 12(4)(1997),59-66.
[37] 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, Vol.18(3)(1997), 49-51.
[38] 王金平, 平面Radon变换的反演公式, CT理论与应用研究, Vol. 24(4)(2000),8-11.
[39] 陈东方, 基于小波变换的Radon反演变换, 安徽大学学报(自然科学版), Vol.24(4)(2000), 24-33.
[40] 尤江生, 包尚联, 完全三维图像重建的一个解析公式, 北京大学学报(自然科学版), Vol. 32(5)(1996), 594-601.
[2] Bloom, W. R. and Heyer, H., Harmonic analysis of probability measures onhypergroup,in ”de Gruyter Studies in Mathematics,”, Vol. 20, de Gruyter,Berlin/New York, (1994).
[3] Deans, S. R., The Radon Transform and some of its application, (New York),Wiley, (1983).
[4] Erdelyi, A., Magnus, W., Oberhettionger, F. and Tricomi, F. G., Higher Tran-scendental functions, Vol. 2, McGraw-Hill New York, (1953).
[5] Faraut, J. and Harzallah, K., Deux cours d’Analyse Harmonique, Bikha¨user,Boston, (1984).
[6] Felix, R., Radon-Transform auf nilpotenten Lie-Gruppen. Invent. Math.,112(1993), 413-443.
[7] Geller, D. and Stein, E. M., Singular convolution operation on the Heisenberggroup, Bull. Amer. Math. Soc., 6(1982), 99-103.
[8] Gonzalez, F. B., Radon transform on grassmann manifolds. J. Funct. Anal.,71(1987), 339-362.
[9] Grossmann, A. and Morlet, J., Decomposition of Hardy functions into squareintegrable wavelets of constant shape, SIAM J. Math. Anal., 15(1984), 723-736.
[10] He, J., A characterization of inverse Radon transform on the Laguerre hyper-group, J. Math. Anal. Appl., 318(2006), 387-395.
[11] He, J., An Inversion Formula of the Radon Transform on the Heisenberg Group,Ganad. Math. Bull. Vol. 47(3)(2004), 389-397.
[12] He, J. and Liu,H., Inversion of the Radon Transform Associated with the Classi-cal Domain of Type One, Int. J. Math., 16(2005), 875-887.
[13] He, J., and Liu, H., Admissible Wavelets and Inversion Radon Transform Asso-ciated With the Affine Homogeneous Siegal Domains of Type II, Comm. Anal.Geom. Vol. 15(1), (2007), 1-28.
[14] He, J. and Liu, H., Admissible wavelets associtated with the affine automorphismgroup of the Siegel upper half-plane, J. Math. Anal. Appl., 208(1997), 58-90.
[15] 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.
[16] Helgason, S., The Radon Transform, 2nd edn. (Birkha¨user, Boston, Basel,Berlin, 1999).
[17] Holschneider, M., Inverse Radon transform through inverse wavelet transform,Inverse Problems, 7(1991), 853-861.
[18] Jewett, R. I., Space with an abstract convolution of measures, Adv. in Math.,18(1975), 1-101.
[19] Koorwinder, T. H., The addition formula for Laguerre Polynomials, SLAM J.Math. Anal., 8(1977), 535-540.
[20] Lin Chincheng, Lp Mulitipliers and Their H1 ? L1 Estimates on the HeisenbergGroup, Revista. Mat. Ibero., 11(1995), 269-308.
[21] Liu, H. and Peng, L., Admissible wavelets associtated with the Heisenberggroup, Pacific J. Math., 180(1997), 101-123.
[22] Ludwig, D., The Radon transform on Euclidean space, Comm. Pure Appl. Math.23(1966), 49-81.
[23] Mauceri, G., Zonal Multipliers on the Heisenberg Group, Pacific J. Math.95(1981), 143-159.
[24] Mu¨ller, D. and Stein, E. M., On Spectral Multiplier for the Heisenberg and Re-lated Groups, J. Math. Pures. Appl, 73(1994), 413-440.
[25] Nessibi, M. M. and Trime`che, K., Inversion of the Radon transform on theLaguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl.,208(1997), 337-363.
[26] Nessibi, M. M., Sifi, M. and Trime`che, K., Continuous wavelet transform andcontinuous multiscale analysis on Laguerre hypergroup, C. R. Math. Rep. Acad.Sci. Canada, 17(1995), 73-78.
[27] Rubin, B., The Caldero′n formula,windowed x-ray transform on the Laguerrehypergroup in Lp-space, J. Fourier Anal. Appl., 4(1998), 175-197.
[28] Rubin, B., Fractional calculus and wavelet transforms in integral geometry,Fractional Calculus and Applied Analysis 1(1998), 193-219.
[29] Solmon, D. C., Asymptotic formulas for the dual Radon transform and applica-tions, Math. Z., 195(1987), 321-343.
[30] Stein, E. M., Harmonic Analysis: Real-Variables Methods, Orthogonality, andOscillatory Integrals, Puinceton University Press, Princeton, (1993).
[31] Stempak, K., Mean summability methods for Laguerre series, Trans. Amer.Math. Soc., 322(1990), 671-690.
[32] Stempak, K., Almost everywhere summability of Laguerre series, Studia Math.,100(1991), 129-147.
[33] Stempak, K., An algebra associated with the generalized sublaplacian, StudiaMath., 88(1988), 245-256.
[34] Strichartz, R. S., Lp Harmonic aralysis and Radon transforms on the Heisenberggroup, J. Funct. Anal., 96(1991), 350-406.
[35] Thangavelu, S. Restriction Theorems for the Heisenberg Group, J. Reine Angew.Math., 414(1991), 51-65.
[36] 李远钦, 刘雯林, n维广义Radon变换, 地球物理学进展, Vol. 12(4)(1997),59-66.
[37] 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, Vol.18(3)(1997), 49-51.
[38] 王金平, 平面Radon变换的反演公式, CT理论与应用研究, Vol. 24(4)(2000),8-11.
[39] 陈东方, 基于小波变换的Radon反演变换, 安徽大学学报(自然科学版), Vol.24(4)(2000), 24-33.
[40] 尤江生, 包尚联, 完全三维图像重建的一个解析公式, 北京大学学报(自然科学版), Vol. 32(5)(1996), 594-601.