2 Nonstandard Sobolev scales and the mapping properties of the X-ray transform on manifolds with strictly convex boundary
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François Monard
Abstract
This article surveys recent results aiming at obtaining refined mapping estimates for the X-ray transform on a Riemannian manifold with boundary, which leverage the condition that the boundary be strictly geodesically convex. These questions are motivated by classical inverse problems questions (e. g., range characterization, stability estimates, mapping properties on Hilbert scales), and more recently by uncertainty quantification and operator learning questions.
Abstract
This article surveys recent results aiming at obtaining refined mapping estimates for the X-ray transform on a Riemannian manifold with boundary, which leverage the condition that the boundary be strictly geodesically convex. These questions are motivated by classical inverse problems questions (e. g., range characterization, stability estimates, mapping properties on Hilbert scales), and more recently by uncertainty quantification and operator learning questions.
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- 1 Generalized Radon transforms 1
- 2 Nonstandard Sobolev scales and the mapping properties of the X-ray transform on manifolds with strictly convex boundary 35
- 3 On geometric inverse problems and microlocal analysis 77
- 4 Inverse problems in cosmological X-ray tomography 139
- 5 The weighted X-ray transform and applications 167
- Index 187
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- 1 Generalized Radon transforms 1
- 2 Nonstandard Sobolev scales and the mapping properties of the X-ray transform on manifolds with strictly convex boundary 35
- 3 On geometric inverse problems and microlocal analysis 77
- 4 Inverse problems in cosmological X-ray tomography 139
- 5 The weighted X-ray transform and applications 167
- Index 187
