Geodesic
Weakly ill-posed problems of integral geometry witch perturbation on polygonal lines
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 1, pp. 5-12.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study a 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. Using these representations 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. 
@article{ISU_2015_15_1_a0,
     author = {A. H. Begmatov and A. O. Pirimbetov and A. K. Seidullaev},
     title = {Weakly ill-posed problems of integral geometry witch perturbation on polygonal lines},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {5--12},
     publisher = {mathdoc},
     volume = {15},
     number = {1},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2015_15_1_a0/}
}
TY  - JOUR
AU  - A. H. Begmatov
AU  - A. O. Pirimbetov
AU  - A. K. Seidullaev
TI  - Weakly ill-posed problems of integral geometry witch perturbation on polygonal lines
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2015
SP  - 5
EP  - 12
VL  - 15
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ISU_2015_15_1_a0/
LA  - ru
ID  - ISU_2015_15_1_a0
ER  - 
%0 Journal Article
%A A. H. Begmatov
%A A. O. Pirimbetov
%A A. K. Seidullaev
%T Weakly ill-posed problems of integral geometry witch perturbation on polygonal lines
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2015
%P 5-12
%V 15
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ISU_2015_15_1_a0/
%G ru
%F ISU_2015_15_1_a0
A. H. Begmatov; A. O. Pirimbetov; A. K. Seidullaev. Weakly ill-posed problems of integral geometry witch perturbation on polygonal lines. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 1, pp. 5-12. http://geodesic.mathdoc.fr/item/ISU_2015_15_1_a0/

[1] Lavrent'ev M. M., Romanov V. G., Shishatskii S. P., Ill-posed problems of mathematical physics and analysis, American Mathematical Society, Providence, 1986, 290 pp.

[2] Natterer F., The mathematics of computerized tomography, Classics in Applied Mathematics, SIAM, Philadelphia, 2001, 222 pp.

[3] Natterer F., Wubbeling F., Mathematical methods in image reconstruction, SIAM, Philadelphia, 2001, 213 pp.

[4] Begmatov A. H., “The integral geometry problem for a family of cones in the $n$-dimensional space”, Siberian Math. J., 37:3 (1996), 430–435 | DOI

[5] Uspenskii S. V., “On reconstruction of a function given integrals over a certain class of conic surfaces”, Siberian Math. J., 18:3 (1977), 675–684 (in Russian)

[6] Begmatov A. H., Pirimbetov A. O., Seidullaev A. K., “Problems of integral geometry in a strip on families of parabolic curves”, Reports of Russian Higher Education Academy of Sciences, 2012, no. 2 (19), 6–15 (in Russian)

[7] Begmatov A. H., Pirimbetov A. O., Seidullaev A. K., “Reconstruction stability in some problems of X-ray and seismic tomography”, Proc. of IFOST-2012, v. II, Tomsk Polytechnic University, 2012, 261–266

[8] Begmatov A. H., Djaykov G. M., “Reconstruction of a function from its spherical means”, Reports of Russian Higher Education Academy of Sciences, 2013, no. 1(20), 6–16 (in Russian)

[9] Begmatov A. H., “The uniqueness of a solution to a Volterra-type integral problem in the plane”, Doklady Math., 80:1 (2009), 528–530 | DOI

[10] Begmatov A. H., “A perturbed integral geometry problem in three-dimensional space”, Siberian Math. J., 41:1 (2000), 1–12 | DOI

[11] Begmatov A. H., Petrova N. N., “The problem of integral geometry with perturbation on elliptic curves in a strip”, Doklady Math., 83:1 (2011), 22–25 | DOI

[12] Trikomi F., Integral equations, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957, 238 pp.