## Abstract

We study a one-parameter family of self-adjoint normal operators for the x-ray transform on the closed Euclidean disk D , obtained by considering specific singularly weighted L ^{2} topologies. We first recover the well-known singular value decompositions in terms of orthogonal disk (or generalized Zernike) polynomials, then prove that each such realization is an isomorphism of C ∞ ( D ) . As corollaries: we give some range characterizations; we show how such choices of normal operators can be expressed as functions of two distinguished differential operators. We also show that the isomorphism property also holds on a class of constant-curvature, circularly symmetric simple surfaces. These results allow to design functional contexts where normal operators built out of the x-ray transform are provably invertible, in Fréchet and Hilbert spaces encoding specific boundary behavior.

Original language | English (US) |
---|---|

Article number | 024001 |

Journal | Inverse Problems |

Volume | 39 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2023 |

## Keywords

- mapping properties
- range characterization
- singular value decomposition
- x-ray transform

## ASJC Scopus subject areas

- Theoretical Computer Science
- Signal Processing
- Mathematical Physics
- Computer Science Applications
- Applied Mathematics