The c∞-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 D, obtained by considering specific singularly weighted L2 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.