Range description for a spherical mean transform on spaces of constant curvature
Let X be a Riemannian manifold and R be the spherical mean transform in X. Let S be a geodesic sphere in X and R S be the restriction of R to the set of geodesic spheres centered on S. We present a complete range description for R S when X is either the hyperbolic space H n or the sphere S n (n ≥ 2 in both cases). The description is analogous to a result for the euclidean space ℝ n obtained by M. Agranovsky, D. Finch, and P. Kuchment and by M. Agranovsky and L. V. Nguyen.
KeywordsInverse Problem Riemannian Manifold Hyperbolic Space Constant Curvature Orthogonality Condition
Unable to display preview. Download preview PDF.
- [AER12]Yuri A. Antipov, Ricardo Estrada, and Boris Rubin, Method of analytic continuation for the inverse spherical mean transform in constant curvature spaces, J. Anal. Math. 118 (2012).Google Scholar
- [Bey83b]Gregory Beylkin, Iterated spherical means in linearized inverse problems, Conference on Inverse Scattering: Theory and Application, SIAM, Philadelphia, PA, 1983, pp. 112–117.Google Scholar
- [Joh81]Fritz John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Springer-Verlag, New York, 1981.Google Scholar
- [LP94]Vladimir Ya. Lin and Allan Pinkus, Approximation of multivariate functions, Advances in Computational Mathematics, World Sci. Publ., River Edge, NJ, 1994, pp. 257–265.Google Scholar
- [Pal10]Victor Palamodov, Remarks on the general Funk transform and thermoacoustic tomography, Inverse Probl. Imaging 4 (2010).Google Scholar
- [Vol02]V. V. Volchkov, Spherical means on symmetric spaces, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 2002, no. 3, 15–19.Google Scholar