Abstract
We study new problem of reconstruction of a function in a strip from their given integrals with known weight function along polygonal lines. We obtained two simply inversion formulas for the solution to the problem. We prove uniqueness and existence theorems for solutions and obtain stability estimates of a solution to the problem in Sobolev's spaces and thus show their weak ill-posedness. Then we consider integral geometry problems with perturbation. The uniqueness theorems are proved and stability estimates of solutions in Sobolev spaces are obtained.
Highlights
The notion of correctness statement of the problem of mathematical physics emerged in the early twentieth century in the writings of eminent French mathematician Jacques Hadamar [1]
There is the problem of solving the operator equation for the function u(x) under the assumption that we are given the right side f ( y), weight function g(x, y) and the variety in which integration is carried out
The problem of solving equation (2) called weakly ill-posed, if for this problem and its solution of the equation, you can to pick up a such pair Function spaces in the definition of the norm involving a finite number of derivatives that the operator handling for this pair of spaces is continuous[2]
Summary
The notion of correctness (correct) statement of the problem of mathematical physics emerged in the early twentieth century in the writings of eminent French mathematician Jacques Hadamar [1]. There is the problem of solving the operator equation for the function u(x) under the assumption that we are given the right side f ( y) , weight function g(x, y) and the variety in which integration is carried out. The problem of solving equation (2) called weakly ill-posed, if for this problem and its solution of the equation, you can to pick up a such pair Function spaces in the definition of the norm involving a finite number of derivatives that the operator handling for this pair of spaces is continuous[2]. If a pair of spaces does not exist, the problem is greatly flawed This classification is the case for the integral geometry problems, and in the general theory of ill-posed problems. The integral geometry problems on the paraboloids with perturbation in three-dimensional layer considered in [13]
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