Helgason's Support Theorem and Spherical Radon Transforms

We prove a new support theorem for the spherical Radon trans- form on manifolds using microlocal analysis, and we discuss the classical ver- sion of this theorem which was proved by Sigurdur Helgason for the spherical transform in R n . We use these theorems and their proofs to find similarities and dierences between the classical and microlocal worlds, and we provide exercises and open problems.

[1]  V. Palamodov,et al.  Reconstruction from limited data of arc means , 2000 .

[2]  Eric Todd Quinto,et al.  Geometry of stationary sets for the wave equation in ℝn: The case of finitely supported initial data , 2001 .

[3]  Carlos A. Berenstein,et al.  Approximation by spherical waves inLp-spaces , 1996 .

[4]  O. Liess.  Conical refraction and higher microlocalization , 1993 .

[5]  Robert S. Strichartz,et al.  RADON INVERSION-VARIATIONS ON A THEME , 1982 .

[6]  E. T. Quinto,et al.  Morera Theorems for Complex Manifolds , 2000 .

[7]  E. T. Quinto,et al.  Local Tomography in Electron Microscopy , 2008, SIAM J. Appl. Math..

[8]  Jan Boman.  An example of non-uniqueness for a generalized Radon transform , 1993 .

[9]  C. Berenstein,et al.  Pompeiu's problem on symmetric spaces , 1980 .

[10]  Akira Kaneko,et al.  Radon transform of hyperfunctions and support theorem , 1995 .

[11]  Jan-Erik Björk,et al.  Rings of differential operators , 1979 .

[12]  Eric Todd Quinto,et al.  Injectivity Sets for the Radon Transform over Circles and Complete Systems of Radial Functions , 1996 .

[13]  S. Helgason.  The surjectivity of invariant di erential operators on symmetric spaces , 1973 .

[14]  Jerrold E. Marsden,et al.  Review: Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, and Nolan R. Wallach, Symplectic geometry and Fourier analysis , 1979 .

[15]  L. Ehrenpreis.  The Universality of the Radon Transform , 2003 .

[16]  M. Cheney,et al.  Synthetic aperture inversion , 2002 .

[17]  S. Helgason.  Support of Radon Transforms , 1980 .

[18]  G. Uhlmann,et al.  Stability estimates for the X-ray transform of tensor fields and boundary rigidity , 2004 .

[19]  Rakesh,et al.  The range of the spherical mean value operator for functions supported in a ball , 2006 .

[20]  E. T. Quinto.  Pompeiu transforms on geodesic spheres in real analytic manifolds , 1993 .

[21]  Sigurdur Helgason,et al.  The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds , 1965 .

[22]  Stationary sets for the wave equation in crystallographic domains , 2003 .

[23]  Eric Todd Quinto,et al.  The invertibility of rotation invariant Radon transforms , 1983 .

[24]  Andrew Béla Frigyik,et al.  The X-Ray Transform for a Generic Family of Curves and Weights , 2007, math/0702065.

[25]  Peter Kuchment,et al.  On the injectivity of the circular Radon transform , 2005 .

[26]  E. Candès,et al.  Curvelets and Fourier Integral Operators , 2003 .

[27]  Peter Kuchment,et al.  A Range Description for the Planar Circular Radon Transform , 2006, SIAM J. Math. Anal..

[28]  C. Berenstein,et al.  A local version of the two-circles theorem , 1986 .

[29]  Akira Kaneko.  Introduction to hyperfunctions , 1988 .

[30]  E. T. Quinto,et al.  Range descriptions for the spherical mean Radon transform. I. Functions supported in a ball , 2006, math/0606314.

[31]  F. John.  Plane Waves and Spherical Means: Applied To Partial Differential Equations , 1981 .

[32]  Rakesh,et al.  Determining a Function from Its Mean Values Over a Family of Spheres , 2004, SIAM J. Math. Anal..

[33]  E. Stein,et al.  Singular integrals related to the Radon transform and boundary value problems. , 1983, Proceedings of the National Academy of Sciences of the United States of America.

[34]  Jan Boman.  A local vanishing theorem for distributions , 1992 .

[35]  Plamen Stefanov,et al.  Recent progress on the boundary rigidity problem , 2005 .

[36]  Eric Todd Quinto,et al.  Singularities of the X-ray transform and limited data tomography , 1993 .

[37]  Jan Boman.  Helgason's support theorem for Radon transforms — A new proof and a generalization , 1991 .

[38]  L. Zalcman.  Offbeat Integral Geometry , 1980 .

[39]  Carlos Alberto Berenstein,et al.  Integral Geometry in Hyperbolic Spaces and Electrical Impedance Tomography , 1996, SIAM J. Appl. Math..

[40]  Classical Microlocal Analysis in the Space of Hyperfunctions , 2000 .

[41]  G. Uhlmann,et al.  Nonlocal inversion formulas for the X-ray transform , 1989 .

[42]  Zero integrals on circles and characterizations of harmonic and analytic functions , 1990 .