ON ONE PROBLEM OF INTEGRAL GEOMETRY

Abstract

The paper is devoted to a problem of integral geometry on a two-dimensional plane. In the band of the two-dimensional space , a two-parameter family of curves is given. Each of these curves consists of two arcs , which are adjacent to each other at the vertex of the curve at an angle , and the other ends rest on the axis. With respect to the branches of the curve , certain smoothness conditions are assumed. The family of curves can be parametrized using the coordinates of the vertices of the curves, that is, each curve from this family can be put in an unambiguous correspondence with the coordinates of its vertex. For each point , there is only one curve belonging to the specified family and having the point as its vertex. It is required to determine the function from the following equation for a given function where – region bounded curves and the axis . The theorem (uniqueness of the solution) is proved. If , then the solution of equation (*) is unique in the class of continuous finite functions with a carrier in . In proving the theorem, the invariance of the family of curves L(ξ,n) and weight functions to shifts along the abscissa axis is significantly used. Further, using the Fourier transform from the function with respect to the variable and the permissible transformations, a family of Volterra integral equations of the second kind is obtained with respect to the Fourier image from the function with respect to the first variable. The paper also provides an estimate of the conditional stability of the considered integral geometry problem