A duality in integral geometry; some generalizations of the Radon transform
Author:
Sigurdur Helgason
Journal:
Bull. Amer. Math. Soc. 70 (1964), 435-446
DOI:
https://doi.org/10.1090/S0002-9904-1964-11147-2
MathSciNet review:
0166795
Full-text PDF Free Access
References | Additional Information
- 1. V. A. Borovikov, Fundamental solutions of linear partial differential equations with constant coefficients, Trudy Moskov. Mat. Obšč. 8 (1959), 199–257 (Russian). MR 0123087
- 2. R. Courant and A. Lax, Remarks on Cauchy’s problem for hyperbolic partial differential equations with constant coefficients in several independent variables, Comm. Pure Appl. Math. 8 (1955), 497–502. MR 75430, https://doi.org/10.1002/cpa.3160080405
- 3. Paul Funk, Über eine geometrische Anwendung der Abelschen Integralgleichung, Math. Ann. 77 (1915), no. 1, 129–135 (German). MR 1511851, https://doi.org/10.1007/BF01456824
- 4. Lars Gȧrding, Transformation de Fourier des distributions homogènes, Bull. Soc. Math. France 89 (1961), 381–428 (French). MR 149195
- 5. I. M. Gel′fand, Integral geometry and its relation to the theory of representations., Russian Math. Surveys 15 (1960), no. 2, 143–151 (Russian). MR 0144358, https://doi.org/10.1070/RM1960v015n02ABEH004218
- 6. I. M. Gel′fand and M. I. Graev, Analogue of the Plancherel formula for the classical groups, Trudy Moskov. Mat. Obšč. 4 (1955), 375–404 (Russian). MR 0071714
- 7. I. M. Gelfand and M. I. Graev, The geometry of homogeneous spaces, group representations in homogeneous spaces and questions in integral geometry related to them. I, Trudy Moscov. Mat. Obšč. 8(1959), 321-390.
- 8. I. M. Gelfand, M. I. Graev and N. Vilenkin, Integral geometry and its relation to problems in the theory of group representations, Vol. 5, Generalized Functions, Fizmatgiz, Moscow, 1962.
- 9. I. M. Gel′fand and M. A. Naĭmark, Unitarnye predstavleniya klassičeskih grupp, Trudy Mat. Inst. Steklov., vol. 36, Izdat. Nauk SSSR, Moscow-Leningrad, 1950 (Russian). MR 0046370
- 10. I. M. Gel′fand and Z. Ya. Šapiro, Homogeneous functions and their extensions, Uspehi Mat. Nauk (N.S.) 10 (1955), no. 3(65), 3–70 (Russian). MR 0073042
- 11. A. A. Hačaturov, Determination of the value of the measure for a region of 𝑛-dimensional Euclidean space from its values for all half-spaces, Uspehi Matem. Nauk (N.S.) 9 (1954), no. 3(61), 205–212 (Russian). MR 0064120
- 12. Harish-Chandra, The Plancherel formula for complex semisimple Lie groups, Trans. Amer. Math. Soc. 76 (1954), 485–528. MR 63376, https://doi.org/10.1090/S0002-9947-1954-0063376-X
- 13. Harish-Chandra, Fourier transforms on a semisimple Lie algebra. I, Amer. J. Math. 79 (1957), 193–257. MR 87044, https://doi.org/10.2307/2372680
- 14. Harish-Chandra, A formula for semisimple Lie groups, Amer. J. Math. 79 (1957), 733–760. MR 96138, https://doi.org/10.2307/2372432
- 15. Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241–310. MR 94407, https://doi.org/10.2307/2372786
- 16. Sigurđur Helgason, Differential operators on homogeneous spaces, Acta Math. 102 (1959), 239–299. MR 117681, https://doi.org/10.1007/BF02564248
- 17. Sigurđur Helgason, Some remarks on the exponential mapping for an affine connection, Math. Scand. 9 (1961), 129–146. MR 131841, https://doi.org/10.7146/math.scand.a-10631
- 18. Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
- 19. S. Helgason, Duality and Radon transform for symmetric spaces, Amer. J. Math. 85 (1963), 667–692. MR 158409, https://doi.org/10.2307/2373114
- 20. S. Helgason, Fundamental solutions of invariant differential operators on symmetric spaces, Bull. Amer. Math. Soc. 69 (1963), 778–781. MR 156919, https://doi.org/10.1090/S0002-9904-1963-11029-0
- 21. Sigurđur Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math. 113 (1965), 153–180. MR 172311, https://doi.org/10.1007/BF02391776
- 22. S. Helgason, The Plancherel formula for the Radon transform on symmetric spaces(to appear).
- 23. Fritz John, Bestimmung einer Funktion aus ihren Integralen Über gewisse Mannigfaltigkeiten, Math. Ann. 109 (1934), no. 1, 488–520 (German). MR 1512906, https://doi.org/10.1007/BF01449151
- 24. Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
- 25. A. A. Kirillov, A problem of I. M. Gel′fand, Soviet Math. Dokl. 2 (1961), 268–269. MR 0117513
- 26. Johann Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Computed tomography (Cincinnati, Ohio, 1982) Proc. Sympos. Appl. Math., vol. 27, Amer. Math. Soc., Providence, R.I., 1982, pp. 71–86 (German). MR 692055
- 27. Georges de Rham, Sur la reductibilité d’un espace de Riemann, Comment. Math. Helv. 26 (1952), 328–344 (French). MR 52177, https://doi.org/10.1007/BF02564308
- 28. Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966 (French). MR 0209834
- 29. V. I. Semjanistyĭ, On some integral transformations in Euclidean space, Dokl. Akad. Nauk SSSR 134 (1960), 536–539 (Russian). MR 0162136
- 30. V. I. Semjanistyĭ, Homogeneous functions and some problems of integral geometry in the spaces of constant curvature, Soviet Math. Dokl. 2 (1961), 59–62. MR 0133006
- 31. Hsien-Chung Wang, Two-point homogeneous spaces, Ann. of Math. (2) 55 (1952), 177–191. MR 47345, https://doi.org/10.2307/1969427
Additional Information
DOI:
https://doi.org/10.1090/S0002-9904-1964-11147-2