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

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.