The Reshetnyak formula and Natterer stability estimates in tensor tomography

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Abstract

The Reshetnyak formula (also known as the Plancherel formula for the Radon transform) states that the Radon transform R is an isometry between and , the latter being the Hilbert space of even functions on furnished by some special norm. We generalize this statement to Sobolev spaces: R is an isometry between and for every real s. Moreover, with the help of Riesz potentials, we define some new Hilbert spaces and prove that R is an isometry between and . The generalized Reshetnyak formula closely relates to the Natterer stability estimates: for functions f supported in a fixed ball. Then we obtain analogs of these statements for the x-ray transform of symmetric tensor fields.

Original languageEnglish
Article number025002
Number of pages20
JournalInverse Problems
Volume33
Issue number2
DOIs
Publication statusPublished - 1 Feb 2017

Keywords

  • Reshetnyak formula
  • stability estimates
  • tensor tomography
  • SPACE

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