Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A Radon transform on spheres through the origin in $ {\bf R}\sp{n}$ and applications to the Darboux equation

Authors: A. M. Cormack and E. T. Quinto
Journal: Trans. Amer. Math. Soc. 260 (1980), 575-581
MSC: Primary 44A05; Secondary 33A45, 35Q05, 43A55, 58G15
MathSciNet review: 574800
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: On domain $ {C^\infty }\,({R^n})$ we invert the Radon transform that maps a function to its mean values on spheres containing the origin. Our inversion formula implies that if $ f\, \in \,{C^\infty }\,({R^n})$ and its transform is zero on spheres inside a disc centered at 0, then f is zero inside that disc. We give functions $ f\, \notin \,{C^\infty }\,({R^n})$ whose transforms are identically zero and we give a necessary condition for a function to be the transform of a rapidly decreasing function. We show that every entire function is the transform of a real analytic function. These results imply that smooth solutions to the classical Darboux equation are determined by the data on any characteristic cone with vertex on the initial surface; if the data is zero near the vertex then so is the solution. If the data is entire then a real analytic solution with that data exists.

References [Enhancements On Off] (What's this?)

  • [1] Y. W. Chen, On the solutions of the wave equation in a quadrant of $ {R^4}$, Bull. Amer. Math. Soc. 70 (1964), 172-177. MR 0161027 (28:4236)
  • [2] A. M. Cormack, Representation of a function by its line integrals with some radiological applications, J. Appl. Phys. 34 (1963), 2722-2727.
  • [3] -, Representation of a function by its line integrals with some radiological applications 11, J. Appl. Phys. 35 (1964), 2908-2913
  • [4] S. R. Deans, A unified Radon inversion formula, J. Math. Phys. 19 (1978), 2346-2349. MR 506707 (80g:44002)
  • [5] J. B. Diaz and H. F. Weinberger, A solution of the singular initial value problem for the Euler-Poisson-Darboux equation, Proc. Amer. Math. Soc. 4 (1953), 703-715. MR 0058099 (15:321g)
  • [6] A. Erdelyi, W. Magnus, R. Oberhettinger and F. Tricomi, Higher transcendental functions, Vol. II, McGraw-Hill, New York, 1953.
  • [7] I. M. Gelfand, M. I. Graev and Z. Ya. Shapiro, Differential forms and integral geometry, Functional Anal. Appl. 3 (1969), 24-40. MR 0244919 (39:6232)
  • [8] I. S. Gradshteyn and I. M. Ryzhik, Tables of integrals series and products, Academic Press, New York, 1965.
  • [9] V. Guillemin and S. Sternberg, Geometric asymptotics, Math. Surveys, no. 14, Amer. Math. Soc., Providence, R. I., 1977. MR 0516965 (58:24404)
  • [10] S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grossman manifolds, Acta Math. 113 (1965), 153-180. MR 0172311 (30:2530)
  • [11] -, The surjectivity of invariant differential operators on symmetric spaces. I, Ann. of Math. 98 (1973), 451-479. MR 0367562 (51:3804)
  • [12] L. Hörmander, An introduction to complex analysis in several variables, Van Nostrand, Princeton, N. J., 1966. MR 0203075 (34:2933)
  • [13] F. John, Plane waves and spherical means applied to partial differential equations, Interscience, New York, 1955. MR 0075429 (17:746d)
  • [14] D. Ludwig, The Radon transform on Euclidean space, Comm. Pure Appl. Math. 69 (1966), 49-81. MR 0190652 (32:8064)
  • [15] R. B. Marr (Ed.), Proceedings of an International Workshop on Three-Dimensional Reconstruction. Brookhaven National Laboratory Technical Report BNL 20425, 1974.
  • [16] S. D. Poisson, Mémoire sur l'intégration de quelques équations linéaires aux différences partielles et particularement de l'équation générale du mouvement des fluides élastique, Mem. Acad. Sci. Inst. France 3 (1818), 121-176.
  • [17] E. T. Quinto, The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc. 257 (1980), 331-346. MR 552261 (81a:58048)
  • [18] J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfälligkeiten, Ber. Verh. Säcks. Akad. 69 (1917), 262-277.
  • [19] H. Rhee, A representation of the solutions of the Darboux equation in odd-dimensional spaces, Trans. Amer. Math. Soc. 150 (1970), 491-498. MR 0262647 (41:7252)
  • [20] R. Seeley, Spherical harmonics, Amer. Math. Monthly 73 (1966), 115-121. MR 0201695 (34:1577)
  • [21] K. T. Smith, D. C. Solmon and S. L. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83 (1977), 1227-1270. MR 0490032 (58:9394a)
  • [22] Alexander Weinstein, On the wave equation and the equation of Euler-Poisson, Proc. Sympos. Appl. Math., vol. 5, McGraw-Hill, New York, 1954, pp. 137-147. MR 0063544 (16:137b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 44A05, 33A45, 35Q05, 43A55, 58G15

Retrieve articles in all journals with MSC: 44A05, 33A45, 35Q05, 43A55, 58G15

Additional Information

Keywords: Radon transform, spherical means, Darboux partial differential equation, Gegenbauer polynomials, spherical harmonics
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society