The $C^{\infty }$-isomorphism property for a class of singularly-weighted X-ray transforms

Abstract

We study a one-parameter family of self-adjoint normal operators for the X-ray transform on the closed Euclidean disk $\mathbb{D}$, 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 $C^{\infty }(\mathbb{D})$. 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.

28 pages, 1 figure

References in corpus (1)

• Sampling the X-ray transform on simple surfaces

Cited by in corpus (2)

• Boundary triples for a family of degenerate elliptic operators of Keldysh type
• Double b-fibrations and desingularization of the X-ray transform on manifolds with strictly convex boundary