ON THE LOWER BOUND IN THE PROBLEM OF APPROXIMATE RECONSTRUCTION OF FUNCTIONS BY VALUES OF THE RADON TRANSFORM

Abstract

In this paper, we study the problem of function reconstruction by values of Radon transforms within the framework of the Computational (Numerical) Diameter (C(N)D) approach. The meaning of C(N)D is to solve two independent problems: obtaining lower bounds of the reconstruction error by exact information and specifying the computing tool that implements the upper bounds (preferably coinciding with the lower bound up to constants). The C(N)D approach is a mathematical model of experiments for describing various processes (physical, chemical, technical, etc.). An important role in setting up such experiments is played by types of measuring instruments, which is reflected in C(N)D as types of functionals. The next important point is the choice of location and balancing of instruments, i.e. selection of functionals’ parameters. The final step is to build an optimal computing tool using the obtained data. The most studied types of functionals are function values at points and Fourier coefficients. An important difference of this work from previously obtained results is the study of the approximation capabilities of another type of functionals – Radon transforms, i.e. mathematical model of the use of tomography and similar technologies. This paper is devoted to obtaining lower bounds for the error in reconstructing functions from Sobolev and Korobov spaces.