Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

TL;DR

This paper establishes higher-order Reshetnyak formulas for the ray transform of symmetric tensor fields on Euclidean space, introducing a computational algorithm for differential operators on the sphere, with explicit formulas for low orders.

Contribution

It develops a general algorithm to derive higher-order Reshetnyak formulas for tensor field transforms, providing explicit results for orders 0, 1, and 2.

Findings

Derived the r-th order Reshetnyak formula for symmetric tensor fields.

Introduced an algorithm to compute differential operators on the sphere.

Presented explicit formulas for low-order cases.

Abstract

For an integer $r\ge0$, we prove the $r$th order Reshetnyak formula for the ray transform of rank $m$ symmetric tensor fields on $\mathbb{R}^n$. Certain differential operators $A^{(m,r,l)}\ (0\le l\le r)$ on the sphere $\mathbb{S}^{n-1}$ are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any $r$ although the volume of calculations grows fast with $r$. The algorithm is realized for small values of $r$ and Reshetnyak formulas of orders $0,1,2$ are presented in an explicit form.