Ray Transforms and Vector Fields

TL;DR

This paper reviews and extends a technique for reconstructing smooth functions from their averages over various curves using complex-analytic methods, providing conditions and stability estimates for different vector fields.

Contribution

It introduces new conditions and extends the complexification approach to more general vector fields for function recovery from curve averages.

Findings

Extended the technique to more general vector fields.

Provided stability estimates for the reconstruction method.

Clarified conditions for the validity of previous formulae.

Abstract

We review and extend a technique for recovering a smooth function from its averages over a wide class of curves in a general region of Euclidean space. The method is based on complexification of the underlying vector fields defining the transport and recasting the problem in terms of complex-analytic function theory. Conditions on the validity of prior formulae appearing in a previous paper as well as stability estimates are then discussed first for the case of vector fields with polynomial coefficients and later for more general cases.