Dynamical approach to some problems in integral geometry

Author:
Boris Paneah

Translated by:

Journal:
Trans. Amer. Math. Soc. **356** (2004), 2757-2780

MSC (2000):
Primary 45E99, 39B22; Secondary 37E99, 35L35

DOI:
https://doi.org/10.1090/S0002-9947-03-03435-4

Published electronically:
December 15, 2003

MathSciNet review:
2052596

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Abstract | References | Similar Articles | Additional Information

Abstract: As is well known, the main problem in integral geometry is to reconstruct a function in a given domain , where its integrals over a family of subdomains in 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 is a bounded domain in 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 with data on the *whole* boundary .

**1.**I.Gel'fand, M.Graev and N.Vilenkin,*Integral Geometry and Representation Theory*, New York, Academic Press, 1966. MR**34:7726****2.**B.Paneah, On a Problem in Integral Geometry Connected to the Dirichlet Problem for Hyperbolic Equations, IMRN International Mathematics Research Notices, 1997, No.5, 213-222. MR**98k:53097****3.**B.Paneah, A Problem in Integral Geometry with Applications to the Dirichlet Problem for a Class of Hyperbolic Differential Equations,*Differential Equations, Asymptotic Analysis and Mathematical Physics*, Math.Res. 100, Akademic-Verlag, Berlin, 1997, 256-261. MR**98h:53115****4.**B.Paneah, On Solvability of Functional Equations Connected with Dynamical Systems with Two Generators, Funct. Anal. Appl. 37, No.1, pp 46-60, 2003.**5.**B.Paneah, Noncommutative Dynamical Systems with Two Generators and Their Applications in Analysis, Discrete and Continuous Dynamical Systems (DCDS), v. 9, No. 6, pp. 1411-1422, 2003.

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Additional Information

**Boris Paneah**

Affiliation:
Department of Mathematics, Technion, Haifa, 32000, Israel

Email:
peter@tx.technion.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-03-03435-4

Received by editor(s):
February 8, 2002

Received by editor(s) in revised form:
January 28, 2003

Published electronically:
December 15, 2003

Article copyright:
© Copyright 2003
American Mathematical Society