The C ∞ -isomorphism property for a class of singularly-weighted x-ray transforms

Rohit Kumar Mishra, François Monard*, Yuzhou Zou

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Article number024001
JournalInverse Problems
Issue number2
StatePublished - Feb 2023


  • 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


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