Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature

2019/06/22 by Mishra, Rohit Kumar, Monard, François
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1906.09389

Abstract

For a one-parameter family of simple metrics of constant curvature (4κ for κ∈ (-1,1)) on the unit disk M, we first make explicit the Pestov-Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also translates into moment conditions \it à la Helgason-Ludwig or Gel'fand-Graev. We then derive an explicit Singular Value Decomposition for the geodesic X-ray transform. Computations dictate a specific choice of weighted L²-L² setting which is equivalent to the L²(M, dVol_κ)→ L²(∂₊ SM, dΣ²) one for any κ∈ (-1,1).

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