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Inversion formula and singularities of the solution for the backprojection operator in tomography

Author: A. G. Ramm
Journal: Proc. Amer. Math. Soc. 124 (1996), 567-577
MSC (1991): Primary 44A15, 45P05
MathSciNet review: 1301044
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Abstract: Let $R^\ast \mu := \int _{S^2} \mu (\alpha , \alpha \cdot x) d\alpha$, $x \in {\mathbb {R}}^n$, be the backprojection operator. The range of this operator as an operator on non-smooth functions $R^\ast : X:=L^\infty _0 (S^{n-1} \times {\mathbb {R}}) \to L_{\mathrm {loc}}^2 ({\mathbb {R}}^n)$ is described and formulas for $(R^\ast )^{-1}$ are derived. It is proved that the operator $R^\ast$ is not injective on $X$ but is injective on the subspace $X_e$ of $X$ which consists of even functions $\mu (\alpha , p) = \mu (-\alpha , -p)$. Singularities of the function $(R^\ast )^{-1} h$ are studied. Here $h$ is a piecewise-smooth compactly supported function. Conditions for $\mu$ to have compact support are given. Some applications are considered.

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A. G. Ramm
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602; Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Keywords: Tomography, range, inversion formulas, backprojection
Received by editor(s): May 10, 1994
Received by editor(s) in revised form: September 12, 1994
Additional Notes: The author thanks NSF and LANL for support, Complutense University, Madrid, for hospitality, and A. Katsevich for discussions.
Communicated by: David Sharp
Article copyright: © Copyright 1996 American Mathematical Society