Opuscula Math. 36, no. 2 (2016), 207-213

Opuscula Mathematica

A two cones support theorem

Ricardo Estrada

Abstract. We show that if the Radon transform of a distribution \(f\) vanishes outside of an acute cone \(C_{0}\), the support of the distribution is contained in the union of \(C_{0}\) and another acute cone \(C_{1}\), the cones are in a suitable position, and \(f\) vanishes distributionally in the direction of the axis of \(C_{1}\), then actually \(\operatorname*{supp}f\subset C_{0}\). We show by examples that this result is sharp.

Keywords: Radon transforms, support theorems, distributions.

Mathematics Subject Classification: 44A12, 46F10.

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  1. D.H. Armitage, M. Goldstein, Nonuniqueness for the Radon transform, Proc. Amer. Math. Soc. 117 (1993), 175-178.
  2. J. Boman, Holmgren's uniqueness theorem and support theorems for real analytic Radon transforms, Contemp. Math. 140 (1992), 23-30.
  3. J. Boman, F. Lindskog, Support theorems for the Radon transform and Cramér-Wold theorems, J. Theor. Probab. 22 (2009), 683-710.
  4. R. Estrada, Vector moment problems for rapidly decreasing functions of several variables, Proc. Amer. Math. Soc. 126 (1998), 761-768.
  5. R. Estrada, Support theorems for Radon transforms of oscillatory distributions, Krugujevac J. Math. 39 (2015), 197-205.
  6. R. Estrada, R.P. Kanwal, A distributional approach to Asymptotics. Theory and Applications, 2nd ed., Birkhäuser, Boston, 2002.
  7. R. Estrada, B. Rubin, Null spaces of Radon transforms, preprint 2015, arXiv:1504.03766.
  8. S. Helgason, The Radon transform in Euclidean spaces, compact two-point homogeneous spaces and Grassman manifolds, Acta Math. 113 (1965), 153-180.
  9. S. Helgason, Geometric Analysis on Symmetric Spaces, Amer. Math. Soc., Providence, 2008.
  10. L. Hörmander, The Analysis of Partial Differential Operators, vol. 1, Distribution Theory and Fourier Analysis, Springer Verlag, Berlin, 1983.
  11. J. Horváth, Topological Vector Spaces and Distributions, vol. I, Addison-Wesley, Reading, Massachusetts, 1966.
  12. D. Ludwig, The Radon transform on Euclidean space, Comm. Pure Appl. Math. 19 (1966), 49-81.
  13. S. Łojasiewicz, Sur la valuer et la limite d'une distribution en un point, Studia Math. 16 (1957), 1-36.
  14. S. Łojasiewicz, Sur la fixation de variables dans une distribution, Studia Math. 17 (1958), 1-64.
  15. A.G. Ramm, Radon transform on distributions, Proc. Japan Acad. 71 (1995), 202-206.
  16. A.G. Ramm, A.I. Katsevich, The Radon Transform and Local Tomography, CRC Press, Boca Raton, 1996.
  17. B. Rubin, Introduction to Radon transforms (with elements of fractional calculus and harmonic analysis), Cambridge University Press, 2015 (to appear).
  18. R.S. Strichartz, Radon inversion - variation on a theme, Am. Math. Mon. 89 (1982), 377-384.
  19. F. Trèves, Topological Vector Spaces, Distributions, and Kernels, Academic Press, New York, 1967.
  20. L. Zalcman, Uniqueness and nonuniqueness for the Radon transform, Bull. London Math. Soc. 14 (1982), 241-245.
  • Ricardo Estrada
  • Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.
  • Communicated by Semyon B. Yakubovich.
  • Received: 2015-07-03.
  • Revised: 2015-09-18.
  • Accepted: 2015-10-11.
  • Published online: 2015-12-18.
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Cite this article as:
Ricardo Estrada, A two cones support theorem, Opuscula Math. 36, no. 2 (2016), 207-213, http://dx.doi.org/10.7494/OpMath.2016.36.2.207

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