## Abstract

The momentum ray transform I
^{k}
integrates a rank m symmetric tensor field f over lines in R
^{n}
with the weight t
^{k}
: (I
^{k}
f)(x,ξ)=∫
^{∞}
_{-∞}
t
^{k}
〈 f(x+tξ),ξ
^{m〉}
dt. In particular, the ray transform I=I0 was studied by several authors since it had many tomographic applications. We present an algorithm for recovering f from the data (I
^{0}
f,I
^{1}
f,…,I
^{m}
f). In the cases of m=1 and m=2, we derive the Reshetnyak formula that expresses ∥f∥Hst(ℝn) through some norm of (I
^{0}
f,I
^{1}
f,…,I
^{m}
f). The Hst-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

Original language | English |
---|---|

Pages (from-to) | 679-701 |

Number of pages | 23 |

Journal | Inverse Problems and Imaging |

Volume | 13 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jun 2019 |

## Keywords

- Inverse problems
- Ray transform
- Reshetnyak formula
- Stability estimates
- Tensor analysis
- tensor analysis
- inverse problems
- stability estimates