A certain inversion problem for the ray transform with incomplete data (Q2760735)

Consider the family of cones \(K(x)=\{\xi\in {\mathbb R}^3:\;|x_3-\xi_3|=|x'-\xi'|\}\), \(x=(x',x_3)\in {\mathbb R}^3\). Under study is the problem of finding a function \(u(x)\) on its integrals over the family of rulings of the cones \(K(x)\). Thus, the author studies the following problem of integral geometry: Find a function \(u(x)\) from the equation NEWLINE\[NEWLINE \int\limits_{{\mathbb R}^1} u(x_1+s\cos\alpha, x_2+s\sin\alpha,x_3+s) dx= f(x,\alpha), \tag{1} NEWLINE\]NEWLINE where the function \(f\) is assumed to be known for \((x,\alpha)\in {\mathbb R}^3\times [0,2\pi]\). The main result of the article is the uniqueness theorem and stability estimates for solutions to this problem. The uniqueness theorem states that a solution to equation (1) is unique in the class of continuous compactly-supported functions.