Reconstruction of a function from known integrals of it that are taken along linear manifolds (Q1098039)

Let the function \(\omega\) (x,s) \((x,s\in R^{n+1})\) have it's support into the cylinder \(K\times R^ 1\) where \(K\subset R^ n\) is the n- dimensional unit ball and \(n\geq 2\). Let \(\omega\) (x,s) have not strong singularities so that for almost all x, \(p\in R^ n\times R^ n\) the function u(x,p) has the meaning \(u(x,p)=^{def}\int^{\infty}_{- \infty}\omega (x+ps,s)ds.\) The problem: To reconstruct the function \(\omega\) from the preassigned values \(\phi (x,p)=u(x,p)|_{x\in \partial K,p\in R^ n}\). The uniqueness theorem for the solution is proved. The inversion formula is found. The problem is reduced to other two-dimensional problem of integral geometry, for which a necessay and sufficient condition for the existence of a solution is found. For the input problem a necessary condition of solvability is established.