Support theorems for totally geodesic Radon transforms on constant curvature spaces. (English) Zbl 0852.44001
Summary: We prove a relation between the \(k\)-dimensional totally geodesic Radon transforms on the various constant curvature spaces using the geodesic correspondence between the spaces. Then we use this relation to obtain improved support theorems for these transforms.
Keywords:
Riemannian manifold; totally geodesic Radon transforms; constant curvature spaces; geodesic correspondence; support theoremsReferences:
| [1] | Carlos A. Berenstein and Enrico Casadio Tarabusi, Range of the \?-dimensional Radon transform in real hyperbolic spaces, Forum Math. 5 (1993), no. 6, 603 – 616. · Zbl 0786.44004 · doi:10.1515/form.1993.5.603 |
| [2] | Jan Boman and Eric Todd Quinto, Support theorems for real-analytic Radon transforms, Duke Math. J. 55 (1987), no. 4, 943 – 948. · Zbl 0645.44001 · doi:10.1215/S0012-7094-87-05547-5 |
| [3] | I. M. Gel\(^{\prime}\)fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. I. M. Gel\(^{\prime}\)fand and G. E. Shilov, Generalized functions. Vol. 2, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1968 [1977]. Spaces of fundamental and generalized functions; Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. I. M. Gel\(^{\prime}\)fand and G. E. Shilov, Generalized functions. Vol. 3, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1967 [1977]. Theory of differential equations; Translated from the Russian by Meinhard E. Mayer. I. M. Gel\(^{\prime}\)fand and N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Applications of harmonic analysis; Translated from the Russian by Amiel Feinstein. I. M. Gel\(^{\prime}\)fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1966 [1977]. Integral geometry and representation theory; Translated from the Russian by Eugene Saletan. |
| [4] | Sigurdur Helgason, Support of Radon transforms, Adv. in Math. 38 (1980), no. 1, 91 – 100. · Zbl 0462.58007 · doi:10.1016/0001-8708(80)90058-4 |
| [5] | -, The Radon transform, Birkhäuser, Boston, 1980. · Zbl 0453.43011 |
| [6] | Sigurdur Helgason, The totally-geodesic Radon transform on constant curvature spaces, Integral geometry and tomography (Arcata, CA, 1989) Contemp. Math., vol. 113, Amer. Math. Soc., Providence, RI, 1990, pp. 141 – 149. · Zbl 0794.44001 · doi:10.1090/conm/113/1108651 |
| [7] | -, Support theorems in integral geometry and their applications, preprint. |
| [8] | Á. Kurusa, The Radon transform on hyperbolic space, Geom. Dedicata 40 (1991), no. 3, 325 – 339. · Zbl 0803.44002 · doi:10.1007/BF00189917 |
| [9] | Árpád Kurusa, The Radon transform on half sphere, Acta Sci. Math. (Szeged) 58 (1993), no. 1-4, 143 – 158. · Zbl 0792.44003 |
| [10] | Á. Kurusa, The invertibility of the Radon transform on abstract rotational manifolds of real type, Math. Scand. 70 (1992), no. 1, 112 – 126. · Zbl 0755.44004 · doi:10.7146/math.scand.a-12389 |
| [11] | Eric Todd Quinto, The invertibility of rotation invariant Radon transforms, J. Math. Anal. Appl. 91 (1983), no. 2, 510 – 522. , https://doi.org/10.1016/0022-247X(83)90165-8 Eric Todd Quinto, Erratum: ”The invertibility of rotation invariant Radon transforms”, J. Math. Anal. Appl. 94 (1983), no. 2, 602 – 603. · Zbl 0517.44010 · doi:10.1016/0022-247X(83)90084-7 |
| [12] | Rolf Schneider, Functions on a sphere with vanishing integrals over certain subspheres., J. Math. Anal. Appl. 26 (1969), 381 – 384. · Zbl 0167.32703 · doi:10.1016/0022-247X(69)90160-7 |
| [13] | Donald C. Solmon, Asymptotic formulas for the dual Radon transform and applications, Math. Z. 195 (1987), no. 3, 321 – 343. · Zbl 0598.44001 · doi:10.1007/BF01161760 |
| [14] | Wu-Yi Hsiang, On the laws of trigonometries of two-point homogeneous spaces, Ann. Global Anal. Geom. 7 (1989), no. 1, 29 – 45. · Zbl 0695.53036 · doi:10.1007/BF00137400 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
