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Spherical mean transform: A PDE approach
1.  Department of Mathematics, University of Idaho, Moscow, Idaho 83844, United States 
We also discuss how the approach works for the hyperbolic and spherical spaces.
References:
[1] 
M. Agranovsky and P. Kuchment, The support theorem for the single radius spherical mean transform, Memoirs on Differential Equations and Mathematical Physics, 52 (2011), 116. 
[2] 
M. Agranovsky and E. Quinto, Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Funct. Anal., 139 (1996), 383414. doi: 10.1006/jfan.1996.0090. 
[3] 
G. Beylkin, The fundamental identity for iterated spherical means and the inversion formula for diffraction tomography and inverse scattering, J. Math. Phys., 24(1983), 13991400. doi: 10.1063/1.525873. 
[4] 
G. Beylkin, Iterated spherical means in linearized inverse problems, in "Conference on Inverse Scattering: Theory and Application" (Tulsa, Okla., 1983), SIAM, Philadelphia, PA, (1983), 112117. 
[5] 
R. Courant and D. Hilbert, "Methods of Mathematical Physics. Vol. II: Partial Differential Equations," (Vol. II by R. Courant), Interscience Publishers, New YorkLondon, 1962. 
[6] 
M. Courdurier, F. Noo, M. Defrise and H. Kudo, Solving the interior problem of computed tomography using a priori knowledge, Inverse problems, 24 (2008), 065001, 27 pp. doi: 10.1088/02665611/24/6/065001. 
[7] 
A. Cormack and E. Quinto, A Radon transform on spheres through the origin in $R^n$ and applications to the Darboux equation, Trans. Amer. Math. Soc., 260 (1980), 575581. doi: 10.2307/1998023. 
[8] 
C. Epstein and B. Kleiner, Spherical means in annular regions, Comm. Pure Appl. Math., 46 (1993), 441451. doi: 10.1002/cpa.3160460307. 
[9] 
D. Finch, S. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 12131240 (electronic). doi: 10.1137/S0036141002417814. 
[10] 
S. Helgason, "Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions," Pure and Applied Mathematics, 113, Academic Press, Inc., Orlando, FL, 1984. 
[11] 
F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations," Reprint of the 1955 original, SpringerVerlag, New YorkBerlin, 1981. 
[12] 
H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography, Physics in Medicine and Biology, 53 (2008), 2207. 
[13] 
V. Lin and A. Pinkus, Fundamentality of ridge functions, J. Approx. Theory, 75 (1993), 295311. doi: 10.1006/jath.1993.1104. 
[14] 
V. Lin and A. Pinkus, Approximation of multivariate functions, in "Advances in Computational Mathematics" (New Delhi, 1993), Ser. Approx. Decompos., 4, World Sci. Publ., River Edge, NJ, (1994), 257265. 
[15] 
L. {Nguyen}, Range description for a spherical mean transform on spaces of constant curvatures, arXiv:1107.1746, (2011). 
show all references
References:
[1] 
M. Agranovsky and P. Kuchment, The support theorem for the single radius spherical mean transform, Memoirs on Differential Equations and Mathematical Physics, 52 (2011), 116. 
[2] 
M. Agranovsky and E. Quinto, Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Funct. Anal., 139 (1996), 383414. doi: 10.1006/jfan.1996.0090. 
[3] 
G. Beylkin, The fundamental identity for iterated spherical means and the inversion formula for diffraction tomography and inverse scattering, J. Math. Phys., 24(1983), 13991400. doi: 10.1063/1.525873. 
[4] 
G. Beylkin, Iterated spherical means in linearized inverse problems, in "Conference on Inverse Scattering: Theory and Application" (Tulsa, Okla., 1983), SIAM, Philadelphia, PA, (1983), 112117. 
[5] 
R. Courant and D. Hilbert, "Methods of Mathematical Physics. Vol. II: Partial Differential Equations," (Vol. II by R. Courant), Interscience Publishers, New YorkLondon, 1962. 
[6] 
M. Courdurier, F. Noo, M. Defrise and H. Kudo, Solving the interior problem of computed tomography using a priori knowledge, Inverse problems, 24 (2008), 065001, 27 pp. doi: 10.1088/02665611/24/6/065001. 
[7] 
A. Cormack and E. Quinto, A Radon transform on spheres through the origin in $R^n$ and applications to the Darboux equation, Trans. Amer. Math. Soc., 260 (1980), 575581. doi: 10.2307/1998023. 
[8] 
C. Epstein and B. Kleiner, Spherical means in annular regions, Comm. Pure Appl. Math., 46 (1993), 441451. doi: 10.1002/cpa.3160460307. 
[9] 
D. Finch, S. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 12131240 (electronic). doi: 10.1137/S0036141002417814. 
[10] 
S. Helgason, "Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions," Pure and Applied Mathematics, 113, Academic Press, Inc., Orlando, FL, 1984. 
[11] 
F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations," Reprint of the 1955 original, SpringerVerlag, New YorkBerlin, 1981. 
[12] 
H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography, Physics in Medicine and Biology, 53 (2008), 2207. 
[13] 
V. Lin and A. Pinkus, Fundamentality of ridge functions, J. Approx. Theory, 75 (1993), 295311. doi: 10.1006/jath.1993.1104. 
[14] 
V. Lin and A. Pinkus, Approximation of multivariate functions, in "Advances in Computational Mathematics" (New Delhi, 1993), Ser. Approx. Decompos., 4, World Sci. Publ., River Edge, NJ, (1994), 257265. 
[15] 
L. {Nguyen}, Range description for a spherical mean transform on spaces of constant curvatures, arXiv:1107.1746, (2011). 
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