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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), 1. Google Scholar 
[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), 383. doi: 10.1006/jfan.1996.0090. Google Scholar 
[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), 1399. doi: 10.1063/1.525873. Google Scholar 
[4] 
G. Beylkin, Iterated spherical means in linearized inverse problems,, in, (1983), 112. Google Scholar 
[5] 
R. Courant and D. Hilbert, "Methods of Mathematical Physics. Vol. II: Partial Differential Equations,", (Vol. II by R. Courant), (1962). Google Scholar 
[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). doi: 10.1088/02665611/24/6/065001. Google Scholar 
[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), 575. doi: 10.2307/1998023. Google Scholar 
[8] 
C. Epstein and B. Kleiner, Spherical means in annular regions,, Comm. Pure Appl. Math., 46 (1993), 441. doi: 10.1002/cpa.3160460307. Google Scholar 
[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), 1213. doi: 10.1137/S0036141002417814. Google Scholar 
[10] 
S. Helgason, "Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions,", Pure and Applied Mathematics, 113 (1984). Google Scholar 
[11] 
F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations,", Reprint of the 1955 original, (1955). Google Scholar 
[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). Google Scholar 
[13] 
V. Lin and A. Pinkus, Fundamentality of ridge functions,, J. Approx. Theory, 75 (1993), 295. doi: 10.1006/jath.1993.1104. Google Scholar 
[14] 
V. Lin and A. Pinkus, Approximation of multivariate functions,, in, 4 (1994), 257. Google Scholar 
[15] 
L. {Nguyen}, Range description for a spherical mean transform on spaces of constant curvatures,, , (2011). Google Scholar 
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), 1. Google Scholar 
[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), 383. doi: 10.1006/jfan.1996.0090. Google Scholar 
[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), 1399. doi: 10.1063/1.525873. Google Scholar 
[4] 
G. Beylkin, Iterated spherical means in linearized inverse problems,, in, (1983), 112. Google Scholar 
[5] 
R. Courant and D. Hilbert, "Methods of Mathematical Physics. Vol. II: Partial Differential Equations,", (Vol. II by R. Courant), (1962). Google Scholar 
[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). doi: 10.1088/02665611/24/6/065001. Google Scholar 
[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), 575. doi: 10.2307/1998023. Google Scholar 
[8] 
C. Epstein and B. Kleiner, Spherical means in annular regions,, Comm. Pure Appl. Math., 46 (1993), 441. doi: 10.1002/cpa.3160460307. Google Scholar 
[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), 1213. doi: 10.1137/S0036141002417814. Google Scholar 
[10] 
S. Helgason, "Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions,", Pure and Applied Mathematics, 113 (1984). Google Scholar 
[11] 
F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations,", Reprint of the 1955 original, (1955). Google Scholar 
[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). Google Scholar 
[13] 
V. Lin and A. Pinkus, Fundamentality of ridge functions,, J. Approx. Theory, 75 (1993), 295. doi: 10.1006/jath.1993.1104. Google Scholar 
[14] 
V. Lin and A. Pinkus, Approximation of multivariate functions,, in, 4 (1994), 257. Google Scholar 
[15] 
L. {Nguyen}, Range description for a spherical mean transform on spaces of constant curvatures,, , (2011). Google Scholar 
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