The Radon transform on the Heisenberg group and the transversal Radon transform

Journal of Functional Analysis - Tập 262 - Trang 234-272 - 2012
B. Rubin1
1Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

Tài liệu tham khảo

Agranovsky, 1993, Morera theorem for holomorphic Hp spaces in the Heisenberg group, J. Reine Angew. Math., 443, 49 Axler, 1992 Berenstein, 1998, Radon transforms, wavelets, and applications, vol. 1684, 1 Berenstein, 2001, Laguerre Calculus and Its Applications on the Heisenberg Group, vol. 22 E.J. Candès, Ridgelets: theory and applications, PhD thesis, Technical Report, Department of Statistics, Stanford University, 1998. Candès, 1999, Harmonic analysis of neural networks, Appl. Comput. Harmon. Anal., 6, 197, 10.1006/acha.1998.0248 Cowling, 1984, Harmonic analysis on Heisenberg type groups from a geometric viewpoint, vol. 1077, 60 Deans, 2007 Ehrenpreis, 2003 Felix, 1993, Radon-Transformation auf nilpotenten Lie-Gruppen, Invent. Math., 112, 413, 10.1007/BF01232441 R. Felix, A general approach to the Radon transform, in: Proceedings of the Conference “Program Systems: Theory and Applications”, Pereslavl-Zalessky, Russia, 2006, 31 pp. Frazier, 1991, Littlewood–Paley Theory and the Study of Function Spaces, vol. 79 Gelfand, 2003, Selected Topics in Integral Geometry, vol. 220 Gelfand, 1966, Generalized Functions, vol. 5 Geller, 1982, Singular convolution operators on the Heisenberg group, Bull. Amer. Math. Soc. (N.S.), 6, 99, 10.1090/S0273-0979-1982-14977-1 Geller, 1984, Estimates for singular convolution operators on the Heisenberg group, Math. Ann., 267, 1, 10.1007/BF01458467 Gradshteyn, 1980 Grafakos, 2008, Classical Fourier Analysis, vol. 249 He, 2004, An inversion formula of the Radon transform on the Heisenberg group, Canad. Math. Bull., 47, 389, 10.4153/CMB-2004-038-9 He, 2006, A characterization of inverse Radon transform on the Laguerre hypergroup, J. Math. Anal. Appl., 318, 387, 10.1016/j.jmaa.2005.05.069 He, 2005, Inversion of the Radon transform associated with the classical domain of type one, Internat. J. Math., 16, 875, 10.1142/S0129167X05003120 He, 2007, Admissible wavelets and inverse Radon transform associated with the affine homogeneous Siegel domains of type II, Comm. Anal. Geom., 15, 1, 10.4310/CAG.2007.v15.n1.a1 Helgason, 1999 Kaplan, 1983, Harmonic analysis on groups of Heisenberg type, vol. 992, 416 Katsevich, 1997, Range of the Radon transform on functions which do not decay fast at infinity, SIAM J. Math. Anal., 28, 852, 10.1137/S0036141095289518 Korányi, 1983, Geometric aspects of analysis on the Heisenberg group, 209 Liu, 2007, Inversion of the Radon transform on the product Laguerre hypergroup by using generalized wavelets, Int. J. Comput. Math., 84, 287, 10.1080/00207160701210927 Lizorkin, 1963, Generalized Liouville differentiation and functional spaces Lpr(En). Imbedding theorems, Mat. Sb., 60, 325 Lizorkin, 1969, Generalized Liouville differentiation and the method of multipliers in the theory of imbeddings of classes of differentiable functions, Proc. Steklov Inst. Math., 105, 105 Lizorkin, 1972, Operators connected with fractional differentiation and classes of differentiable functions, Proc. Steklov Inst. Math., 117, 251 Markoe, 2006, Analytic Tomography, vol. 106 Mikhlin, 1970 Müller, 1995, Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. I, Invent. Math., 119, 199, 10.1007/BF01245180 Natterer, 1986 Nessibi, 1997, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl., 208, 337, 10.1006/jmaa.1997.5299 Nogin, 1999, Some applications of potentials and approximative inverse operators in multi-dimensional fractional calculus, Fract. Calc. Appl. Anal., 2, 205 Oberlin, 1982, Mapping properties of the Radon transform, Indiana Univ. Math. J., 31, 641, 10.1512/iumj.1982.31.31046 Ournycheva, 2008, Semyanistyiʼs integrals and Radon transforms on matrix spaces, J. Fourier Anal. Appl., 14, 60, 10.1007/s00041-007-9002-0 Palamodov, 2004, Reconstructive Integral Geometry, vol. 98 Peng, 2007, Radon transform on H-type and Siegel-type nilpotent groups, Internat. J. Math., 18, 1061, 10.1142/S0129167X07004412 1994, vol. 30 Radon, 1917, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math., Nat. Kl., 69, 262 Ramm, 1996 Rouvière, 2001, Inverting Radon transforms: The group-theoretic approach, Enseign. Math., 47, 205 Rubin, 1996, Fractional Integrals and Potentials, vol. 82 Rubin, 1998, Fractional calculus and wavelet transforms in integral geometry, Fract. Calc. Appl. Anal., 1, 193 Rubin, 1998, Spherical Radon transform and related wavelet transforms, Appl. Comput. Harmon. Anal., 5, 202, 10.1006/acha.1997.0228 Rubin, 2002, Inversion formulas for the spherical Radon transform and the generalized cosine transform, Adv. in Appl. Math., 29, 471, 10.1016/S0196-8858(02)00028-3 Rubin, 2002, Radon, cosine, and sine transforms on real hyperbolic space, Adv. Math., 170, 206, 10.1016/S0001-8708(02)92074-8 Rubin, 2003, Notes on Radon transforms in integral geometry, Fract. Calc. Appl. Anal., 6, 25 Rubin, 2004, Reconstruction of functions from their integrals over k-planes, Israel J. Math., 141, 93, 10.1007/BF02772213 Rubin, 2004, Convolution-backprojection method for the k-plane transform, and Calderónʼs identity for ridgelet transforms, Appl. Comput. Harmon. Anal., 16, 231, 10.1016/j.acha.2004.03.003 Rubin, 2004, Generalizations of the Busemann–Petty problem for sections of convex bodies, J. Funct. Anal., 213, 473, 10.1016/j.jfa.2003.10.008 Samko, 1977, Test functions vanishing on a given set, and division by a function, Mat. Zametki, 21, 677 Samko, 1982, Denseness of Lizorkin-type spaces ΦV in Lp(Rn), Mat. Zametki, 31, 655 Samko, 2002, Hypersingular Integrals and Their Applications, vol. 5 Samko, 1993 Samko, 1999, On approximate definition of fractional differentiation, Fract. Calc. Appl. Anal., 2, 329 Semyanistyi, 1960, On some integral transformations in Euclidean space, Dokl. Akad. Nauk SSSR, 134, 536 Semyanistyi, 1961, Homogeneous functions and some problems of integral geometry in spaces of constant curvature, Sov. Math. Dokl., 2, 59 Smith, 1997, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc., 83, 1227, 10.1090/S0002-9904-1977-14406-6 Solmon, 1979, A note on k-plane integral transforms, J. Math. Anal. Appl., 71, 351, 10.1016/0022-247X(79)90196-3 Stein, 1970 Stein, 1993 Strichartz, 1981, Lp-estimates for Radon transforms in Euclidean and non-Euclidean spaces, Duke Math. J., 48, 699, 10.1215/S0012-7094-81-04839-0 Strichartz, 1991, Lp harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal., 96, 350, 10.1016/0022-1236(91)90066-E Thangavelu, 1998, Harmonic Analysis on the Heisenberg Group, vol. 159 Vladimirov, 1988