Pradipta Chatterjee | Searcharxiv

arXiv2026

A Unified Range Characterization for the Spherical mean transform

In this article, we investigate the range characterization for the spherical mean transform (SMT) of functions supported in the unit ball. In earlier works, in the case of odd dimensions, a set of differential conditions was obtained, whereas in the case of even dimensions, integral conditions were obtained. We prove that these conditions that are different based on the parity of dimension are, in fact, equivalent in odd dimensions. This equivalence shows that the integral conditions yield a unified simple range characterization for the SMT that is valid in both even and odd dimensions.

math.AP↗

arXiv2026

Explicit inversion of spherical Radon transforms in odd dimensions with partial radial data

We derive an explicit inversion algorithm for the spherical Radon transform in odd dimensions with partial radial data. We prove that the reconstruction of the unknown function can be reduced to solving ordinary differential equations, thereby providing a more explicit approach in odd dimensions than solving Volterra integral equation of the first kind established in prior works. We also provide analytical solutions in some special cases. Finally, we present numerical simulations validating our theoretical results. Our work answers a question posed by Rubin in ``Inversion formulae for the spherical mean in odd dimensions and the Euler-Poisson-Darboux equation,'' Inverse Problems 24 (2008), no. 2, 025021, 10 pp.

math.AP↗