The C∞-isomorphism property for a class of singularly-weighted x-ray transforms. (19th December 2022)

• Record Type: Journal Article
• Authors: Mishra, Rohit Kumar; Monard, François; Zou, Yuzhou
• Is Part Of: Inverse problems. Volume 39:Number 2(2023)
• Publication Date: 2022-12-19
• Subjects: singular value decomposition -- range characterization -- mapping properties -- x-ray transform; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aca8cb
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: 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.