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Method of analytic continuation for the inverse spherical mean transform in constant curvature spaces

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Abstract

The following problem arises in thermoacoustic tomography and has intimate connection with PDEs and integral geometry. Reconstruct a function f supported in an n-dimensional ball B given the spherical means of f over all geodesic spheres centered on the boundary of B. We propose a new approach to this problem, which yields explicit reconstruction formulas in arbitrary constant curvature space, including euclidean space ℝn, the n-dimensional sphere, and hyperbolic space. The main idea is analytic continuation of the corresponding operator families. The results are applied to inverse problems for a large class of Euler-Poisson-Darboux equations in constant curvature spaces of arbitrary dimension.

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Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Yuri A. Antipov, Ricardo Estrada & Boris Rubin

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  1. Yuri A. Antipov
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  2. Ricardo Estrada
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  3. Boris Rubin
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Correspondence to Yuri A. Antipov.

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Supported by the NSF grant DMS-0707724.

Supported by the NSF grant PHYS-0968448.

Supported in part by the NSF grant DMS-0556157 and the Louisiana EPSCoR program, and sponsored by NSF, the Board of Regents Support Fund, and the Hebrew University of Jerusalem.

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Antipov, Y.A., Estrada, R. & Rubin, B. Method of analytic continuation for the inverse spherical mean transform in constant curvature spaces. JAMA 118, 623–656 (2012). https://doi.org/10.1007/s11854-012-0046-y

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