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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Inversion formula and singularities of the solution for the backprojection operator in tomography
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by A. G. Ramm PDF
Proc. Amer. Math. Soc. 124 (1996), 567-577 Request permission

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.
References
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Additional Information
  • A. G. Ramm
  • Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602; Los Alamos National Laboratory, Los Alamos, New Mexico 87545
  • Email: ramm@math.ksu.edu
  • 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
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 567-577
  • MSC (1991): Primary 44A15, 45P05
  • DOI: https://doi.org/10.1090/S0002-9939-96-03155-3
  • MathSciNet review: 1301044