Dynamical approach to some problems in integral geometry

Abstract

As is well known, the main problem in integral geometry is to reconstruct a function in a given domain D D , where its integrals over a family of subdomains in D D are known. Such a problem is interesting not only as an object of pure analysis, but also in connection with various applications in practical disciplines. The most remarkable example of such a connection is the Radon problem and tomography. In this paper we solve one of these problems when D D is a bounded domain in R 2 {\mathbb {R}}^2 with a piecewise smooth boundary. Some intermediate results related to dynamical systems with two generators and to some functional-integral equations are new and interesting per se. As an application of the results obtained we briefly study a boundary problem for a general third order hyperbolic partial differential equation in a bounded domain D ⊂ R 2 D\subset {\mathbb {R}}^2 with data on the whole boundary ∂ D \partial D .