The Inverse Problem for the Euler-Poisson-Darboux Equation and Shifted $k$-Plane Transforms

TL;DR

This paper investigates the inverse problem for the Euler-Poisson-Darboux equation, linking it to the injectivity of shifted k-plane transforms and exploring related generalizations like Radon transforms over strips and tubes.

Contribution

It establishes new connections between the inverse Euler-Poisson-Darboux problem and shifted k-plane transform injectivity, including generalizations to Radon transforms over strips and tubes.

Findings

Proved injectivity conditions for shifted k-plane transforms.

Extended analysis to Radon transforms over strips and tubes.

Provided new insights into reconstructing solutions from incomplete data.

Abstract

The inverse problem for the Euler-Poisson-Darboux equation deals with reconstruction of the Cauchy data for this equation from incomplete information about its solution. In the present article, this problem is studied in connection with the injectivity of the shifted $k$-plane transform, which assigns to functions in $L^p(\mathbb {R}^n)$ their mean values over all k-planes at a fixed distance from the given $k$-planes. Several generalizations, including the Radon transform over strips of fixed width in $\mathbb {R}^2$ and a similar transform over tubes of fixed diameter in $\mathbb {R}^3$, are considered.