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2012 | OriginalPaper | Chapter

On Local Injectivity for Generalized Radon Transforms

Author : Jan Boman

Published in: The Mathematical Legacy of Leon Ehrenpreis

Publisher: Springer Milan

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Abstract

We consider a class of weighted plane generalized Radon transforms Rf(γ)=∫f(x,u(ξ,η,x))m(ξ,η,x) dx, where the curve γ=γ _(ξ,η) is defined by y=u(ξ,η,x), and m(ξ,η,x) is a given positive weight function. We prove local injectivity for this transform across a given curve γ ⁰ near a given point (x ⁰,y ⁰) on γ ⁰ for classes of curves and weight functions that are invariant under arbitrary smooth coordinate transformations in the plane.

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Arnold, V.I.: Geometric Methods in the Theory of Ordinary Differential Equations, 2nd edn. Springer, Berlin (1988) CrossRef

Boman, J.: An example of non-uniqueness for a generalized Radon transform. J. Anal. Math. 61, 395–401 (1993) MathSciNetMATHCrossRef

Boman, J.: A local uniqueness theorem for weighted Radon transforms. Inverse Probl. Imaging 4, 631–637 (2010) MathSciNetMATHCrossRef

Boman, J.: Local non-injectivity for weighted Radon transforms. Contemp. Math. 559, 39–47 (2011) MathSciNetCrossRef

Boman, J., Quinto, E.T.: Support theorems for real-analytic Radon transforms. Duke Math. J. 55, 943–948 (1987) MathSciNetMATHCrossRef

Bryant, R., Griffiths, P., Hsu, L.: Toward a geometry of differential equations. In: Geometry, Topology, & Physics, Conf. Proc. Lecture Notes Geom. Topology, vol. IV. Int. Press, Cambridge (1995)

Ehrenpreis, L.: The Universality of the Radon Transform. Clarendon Press, Oxford (2003) MATHCrossRef

Gelfand, I.M., Graev, M.I.: Admissible n-dimensional complexes of curves in R ⁿ. Funct. Anal. Appl. 14, 274–281 (1980) MathSciNetCrossRef

Gelfand, I.M., Graev, M.I., Shapiro, Z.Ya.: Differential forms and integral geometry. Funct. Anal. Appl. 3, 101–114 (1969) CrossRef

10.

Gelfand, I.M., Gindikin, S.G., Shapiro, Z.Ya.: A local problem of integral geometry in a space of curves. Funct. Anal. Appl. 13, 87–102 (1979) MathSciNetCrossRef

11.

Gindikin, S.G.: Reduction of manifolds of rational curves and related problems of the theory of differential equations. Funct. Anal. Appl. 18, 278–298 (1984) MATHCrossRef

12.

Gindikin, S.G.: A remark on the weighted Radon transform on the plane. Inverse Probl. Imaging 4, 649–653 (2010) MathSciNetMATHCrossRef

13.

Gindikin, S.G.: Remarks on infinitesimally Desarguian families of curves. Funct. Anal. Appl. 45, 265–270 (2011) MathSciNetCrossRef

14.

Helgason, S.: A duality in integral geometry on symmetric spaces. In: Proc. U.S.–Japan Seminar on Differential Geometry, Kyoto 1965, pp. 37–56. Nippon Hyoronsha, Tokyo (1966)

15.

Strichartz, R.S.: Radon inversion—variations on a theme. Am. Math. Mon. 89, 377–384 (1982), see also 420–423 MathSciNetMATHCrossRef

Title: On Local Injectivity for Generalized Radon Transforms
Author: Jan Boman
Publisher: Springer Milan
Book: The Mathematical Legacy of Leon Ehrenpreis
Print ISBN: 978-88-470-1946-1

Electronic ISBN: 978-88-470-1947-8

Copyright Year: 2012
DOI: https://doi.org/10.1007/978-88-470-1947-8_5

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