Dynamical approach to some problems in integral geometry

Paneah, Boris

doi:10.1090/S0002-9947-03-03435-4

Dynamical approach to some problems in integral geometry
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Trans. Amer. Math. Soc. 356 (2004), 2757-2780 Request permission

Abstract:

As is well known, the main problem in integral geometry is to reconstruct a function in a given domain $D$, where its integrals over a family of subdomains in $D$ 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 $D$ is a bounded domain in ${\mathbb {R}}^2$ 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 $D\subset {\mathbb {R}}^2$ with data on the whole boundary $\partial D$.

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