Radon transforms over lower-dimensional horospheres in real hyperbolic space
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- by William O. Bray and Boris Rubin PDF
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Abstract:
We study horospherical Radon transforms that integrate functions on the $n$-dimensional real hyperbolic space over horospheres of arbitrary fixed dimension $1\le d\le n-1$. Exact existence conditions and new explicit inversion formulas are obtained for these transforms acting on smooth functions and functions belonging to $L^p$. The case $d=n-1$ agrees with the well-known Gelfand-Graev transform.References
- Carlos A. Berenstein and Enrico Casadio Tarabusi, An inversion formula for the horocyclic Radon transform on the real hyperbolic space, Tomography, impedance imaging, and integral geometry (South Hadley, MA, 1993) Lectures in Appl. Math., vol. 30, Amer. Math. Soc., Providence, RI, 1994, pp. 1â6. MR 1297561
- William O. Bray, Aspects of harmonic analysis on real hyperbolic space, Fourier analysis (Orono, ME, 1992) Lecture Notes in Pure and Appl. Math., vol. 157, Dekker, New York, 1994, pp. 77â102. MR 1277819
- William O. Bray and Boris Rubin, Inversion of the horocycle transform on real hyperbolic spaces via a wavelet-like transform, Analysis of divergence (Orono, ME, 1997) Appl. Numer. Harmon. Anal., BirkhĂ€user Boston, Boston, MA, 1999, pp. 87â105. MR 1731261
- Bent Fuglede, An integral formula, Math. Scand. 6 (1958), 207â212. MR 105724, DOI 10.7146/math.scand.a-10545
- I. M. Gelâfand, Integral geometry and its relation to the theory of representations, Russian Math. Surveys 15 (1960), no. 2, 143â151.
- I. M. GelâČfand and M. I. Graev, Geometry of homogeneous spaces, representations of groups in homogeneous spaces and related questions of integral geometry. I, Trudy Moskov. Mat. ObĆĄÄ. 8 (1959), 321â390; addendum 9 (1959), 562 (Russian). MR 0126719
- I. M. GelâČfand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5: Integral geometry and representation theory, Academic Press, New York-London, 1966. Translated from the Russian by Eugene Saletan. MR 0207913
- Simon Gindikin, Integral geometry on hyperbolic spaces, Harmonic analysis and integral geometry (Safi, 1998) Chapman & Hall/CRC Res. Notes Math., vol. 422, Chapman & Hall/CRC, Boca Raton, FL, 2001, pp. 41â46. MR 1789143
- S. G. Gindikin, Horospherical transform on symmetric Riemannian manifolds of noncompact type, Funktsional. Anal. i Prilozhen. 42 (2008), no. 4, 50â59, 111â112 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 42 (2008), no. 4, 290â297. MR 2492426, DOI 10.1007/s10688-008-0042-2
- Simon Gindikin, Local inversion formulas for horospherical transforms, Mosc. Math. J. 13 (2013), no. 2, 267â280, 363 (English, with English and Russian summaries). MR 3134907, DOI 10.17323/1609-4514-2013-13-2-267-280
- Fulton B. Gonzalez, Conical distributions on the space of flat horocycles, J. Lie Theory 20 (2010), no. 3, 409â436. MR 2743098
- Fulton Gonzalez and Eric Todd Quinto, Support theorems for Radon transforms on higher rank symmetric spaces, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1045â1052. MR 1205492, DOI 10.1090/S0002-9939-1994-1205492-3
- Sigurdur Helgason, The surjectivity of invariant differential operators on symmetric spaces. I, Ann. of Math. (2) 98 (1973), 451â479. MR 367562, DOI 10.2307/1970914
- Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
- Sigurdur Helgason, Geometric analysis on symmetric spaces, 2nd ed., Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 2008. MR 2463854, DOI 10.1090/surv/039
- Sigurdur Helgason, Integral geometry and Radon transforms, Springer, New York, 2011. MR 2743116, DOI 10.1007/978-1-4419-6055-9
- Sigurdur Helgason, Support theorems for horocycles on hyperbolic spaces, Pure Appl. Math. Q. 8 (2012), no. 4, 921â927. MR 2959914, DOI 10.4310/PAMQ.2012.v8.n4.a4
- J. Hilgert, A. Pasquale, and E. B. Vinberg, The dual horospherical Radon transform for polynomials, Mosc. Math. J. 2 (2002), no. 1, 113â126, 199 (English, with English and Russian summaries). MR 1900587, DOI 10.17323/1609-4514-2002-2-1-113-126
- J. Hilgert, A. Pasquale, and E. B. Vinberg, The dual horospherical Radon transform as a limit of spherical Radon transforms, Lie groups and symmetric spaces, Amer. Math. Soc. Transl. Ser. 2, vol. 210, Amer. Math. Soc., Providence, RI, 2003, pp. 135â143. MR 2018358, DOI 10.1090/trans2/210/10
- E. Kelly, Lower-dimensional horocycles and the Radon transforms on symmetric spaces, Preprint, 1974.
- P. I. Lizorkin, Direct and inverse theorems in approximation theory for functions on a LobachevskiÄ space, Trudy Mat. Inst. Steklov. 194 (1992), no. Issled. po Teor. Differ. FunktsiÄ Mnogikh Peremen. i ee Prilozh. 14, 120â147 (Russian); English transl., Proc. Steklov Inst. Math. 4(194) (1993), 125â151. MR 1289652
- Mitsuo Morimoto, Sur les transformations horosphĂ©riques gĂ©nĂ©ralisĂ©es dans les espaces homogĂšnes, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 65â83 (1966) (French). MR 209407
- Mitsuo Morimoto, On the Radon transform, SĆ«gaku 20 (1968), 1â7 (Japanese). MR 252570
- Boris Rubin, Helgason-Marchaud inversion formulas for Radon transforms, Proc. Amer. Math. Soc. 130 (2002), no. 10, 3017â3023. MR 1908925, DOI 10.1090/S0002-9939-02-06554-1
- Boris Rubin, Inversion formulas for the spherical Radon transform and the generalized cosine transform, Adv. in Appl. Math. 29 (2002), no. 3, 471â497. MR 1942635, DOI 10.1016/S0196-8858(02)00028-3
- Boris Rubin, Radon, cosine and sine transforms on real hyperbolic space, Adv. Math. 170 (2002), no. 2, 206â223. MR 1932329, DOI 10.1006/aima.2002.2074
- Boris Rubin, On the Funk-Radon-Helgason inversion method in integral geometry, Geometric analysis, mathematical relativity, and nonlinear partial differential equations, Contemp. Math., vol. 599, Amer. Math. Soc., Providence, RI, 2013, pp. 175â198. MR 3202479, DOI 10.1090/conm/599/11908
- Boris Rubin, Introduction to Radon transforms, Encyclopedia of Mathematics and its Applications, vol. 160, Cambridge University Press, New York, 2015. With elements of fractional calculus and harmonic analysis. MR 3410931
- B. Rubin, Overdetermined transforms in integral geometry, Complex analysis and dynamical systems VI. Part 1, Contemp. Math., vol. 653, Amer. Math. Soc., Providence, RI, 2015, pp. 291â313. MR 3453082, DOI 10.1090/conm/653/13200
- Boris Rubin, New inversion formulas for the horospherical transform, J. Geom. Anal. 27 (2017), no. 1, 908â946. MR 3606575, DOI 10.1007/s12220-016-9704-0
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- N. Ja. Vilenkin, Special functions and the theory of group representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by V. N. Singh. MR 0229863
- N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions. Vol. 2, Mathematics and its Applications (Soviet Series), vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1993. Class I representations, special functions, and integral transforms; Translated from the Russian by V. A. Groza and A. A. Groza. MR 1220225, DOI 10.1007/978-94-017-2883-6
Additional Information
- William O. Bray
- Affiliation: Department of Mathematics, Missouri State University, Springfield, Missouri 65897
- MR Author ID: 189820
- Email: wbray@missouristate.edu
- Boris Rubin
- Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
- MR Author ID: 209987
- Email: borisr@math.lsu.edu
- Received by editor(s): July 18, 2017
- Received by editor(s) in revised form: April 19, 2018
- Published electronically: April 4, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 1091-1112
- MSC (2010): Primary 44A12; Secondary 44A15
- DOI: https://doi.org/10.1090/tran/7666
- MathSciNet review: 3968796