The Problem of Integral Geometry of Volterra Type with a Weight Function of a Special Type

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.


Introduction
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]. Problems of mathematical physic so boundary problem for equations with partial derivatives called correctly if the following conditions: 1. The solution of the problem exists; 2. The problem is unique; 3. The solution depends continuously on the data of the problem.
When studying the mathematical models of technical tasks naturally reformulate specified conditions in the following form: A. Solution task exists for all data belonging to some closed space in nor med linear spaces of type l p p p k W I L C , , , and belongs to the same type; B. Solution the task is only in any analogous space; C. Infinitely small variations of these tasks in the data space, correspond to infinitely small in the solution space, variations of the solution The classic example the Tasks, in correctly to Hadamard, is the Cauchy problem for the Laplace equation.
Operator equation of the first kind is called the equation where x , f -elements of the spaces − F A X , , compact operator from X to F . The operator 1 − A is not continuous-the problem of solving equation (1) is incorrect.
The right side of equation (1) is often obtained on the basis of evidence of physical devices. Norm in the space corresponds to the fact, that is known to estimate the maximum measurement error. The norm in 1 L represents the mean square error.
The integral geometry problem (2) There is the problem of solving the operator equation for the function ) (x u under the assumption that we are given the right side ) ( y f , weight function ) , ( y x g and the variety in which integration is carried out.
The main objective of computed tomographyis an integral geometry problem.
Let − ) (x u the coefficient of absorption; − I radiation intensity. Then Accordingly, the initial intensity of beam L , and its intensity after passing through the body.
Integrating this expression, we obtain 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, then the problem is greatly flawed. Of course, this classification is the case not only for the integral geometry problems, but also in the general theory of ill-posed problems.
V.G. Romanov in [5] investigated the questions of uniqueness and stability of solution of integral geometry in the case where the manifold on which integration is carried out, have the kind of paraboloids, weighting functions and diversity are invariant under the group of all the movements along a fixed hyperplane.
The integral geometry problems on the paraboloids with perturbation in three-dimensional layer considered in [13].

Recovering a Function from Polygonal Lines
Let G -set of bounded functions ) (⋅ g , defined on 1 R and satisfy the following conditions: i.e. С=N, n=1, m=2, p=0, g∈ G . We introduce the notation: that is uniquely parameterized by the coordinates of its vertices where g -weight function of the set G. Function ) , ( y x u -is function of class U, which has all continuous partial derivative sup to − + + ) 4 ( m n the second order inclusive and finite with the carrier and The following theorem holds:

Then the solution of1in the class U is unique and has the representation
Proof. For the first integral of equation (7) we introduce the following change of variables: and for the second term of the left side of (7), we apply the following change: As a result, going to (7) to integration over , where . η − = y h Applying the Fourier transform to the variable x to both sides of equation (11) we obtain: has a zero growth rate. This fact follows from the finiteness ) , ( y x u and mean value theorem. Thus, ) , ( y f λ is a function of the original. Therefore according to Theorem 1(see [4]) to both sides of equation (7), as well as the equation (12)can be used the Laplace transform in y.
Applicable to the equation (12)the Laplace Transform in the variable y. Using Fubini's theorem, we obtain:  Using (9) and (17)

Uniqueness and Stability Problems with Indignation
Now examine the problem of integral geometry with indignation.

Through ( , )
S x y denote the part 2 R + , limited polyline ( , ) P x y and axis 0 . y = Ω there is a stripe: x η ∈ Ω × Ω together