Geodesic
On the regularization of an equation of the first kind with a multiple integration operator
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 4, pp. 578-586 
S. Yu. Sovetnikova; G. V. Khromova. On the regularization of an equation of the first kind with a multiple integration operator. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 4, pp. 578-586. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_4_a1/
@article{ZVMMF_2007_47_4_a1,
     author = {S. Yu. Sovetnikova and G. V. Khromova},
     title = {On the regularization of an equation of the first kind with a~multiple integration operator},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {578--586},
     year = {2007},
     volume = {47},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_4_a1/}
}
TY  - JOUR
AU  - S. Yu. Sovetnikova
AU  - G. V. Khromova
TI  - On the regularization of an equation of the first kind with a multiple integration operator
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2007
SP  - 578
EP  - 586
VL  - 47
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_4_a1/
LA  - ru
ID  - ZVMMF_2007_47_4_a1
ER  - 
%0 Journal Article
%A S. Yu. Sovetnikova
%A G. V. Khromova
%T On the regularization of an equation of the first kind with a multiple integration operator
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2007
%P 578-586
%V 47
%N 4
%U http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_4_a1/
%G ru
%F ZVMMF_2007_47_4_a1

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

The zero-order Tikhonov regularization method as applied to an equation of the first kind with a multiple differentiation operator is considered for the case when the solution belongs to a class from the domain of the adjoint operator. An estimate of the error of the approximate solution in the uniform metric is obtained, which is sharp with respect to the order, and the order is established. It is proved that the proposed method is optimal with respect to the order. Unimprovable estimates of the order of the modulus of continuity of the inverse operator are obtained.

[1] Tikhonov A. N., “O regulyarizatsii nekorrektno postavlennykh zadach”, Dokl. AN SSSR, 153:1 (1963), 49–52 | MR | Zbl

[2] Khudak Yu. I., “O regulyarizatsii reshenii uravnenii I roda”, Zh. vychisl. matem. i matem. fiz., 6:4 (1966), 766–769

[3] Khromova G. V., “O nakhozhdenii ravnomernykh priblizhenii k resheniyu integralnykh uravnenii I roda”, Differents. ur-niya i vychisl. matem., Sb. nauch. tr., v. 4, Izd-vo Saratovskogo un-ta, Saratov, 1974, 3–10

[4] Bakushinskii A. B., Goncharskii A. B., Nekorrektnye zadachi. Chislennye metody i prilozheniya, MGU, M., 1989

[5] Strakhov V. N., “O postroenii optimalnykh po poryadku priblizhennykh reshenii uslovno-korrektnykh zadach”, Differents. ur-niya, 9:10 (1973), 1862–1874

[6] Khromova G. V., “Ob otsenkakh pogreshnostei priblizhennykh reshenii uravnenii I roda”, Dokl. RAN, 378:5 (2001), 605–609 | MR | Zbl

[7] Khromova G. V., “O modulyakh nepreryvnosti neogranichennykh operatorov i optimalnosti metodov priblizhennogo resheniya uravnenii I roda”, Matem. Mekhan., 7, Izd-vo Saratovskogo un-ta, Saratov, 2005, 132–134

[8] Sovetnikova S. Yu., “O skorosti approksimatsii resheniya odnogo integralnogo uravneniya I roda”, Teoriya funktsii, ee prilozheniya i smezhnye voprosy, Tezisy dokl. VII Mezhdunar. kazanskoi letnei shkoly-konf., 30, Izd-vo Kazanskogo un-ta, Kazan, 2005, 144–145

[9] Sovetnikova S. Yu., “O tochnom poryadke skorosti approksimatsii reshenii odnogo klassa integralnykh uravnenii I roda”, Sovrem. probl. teorii funktsii i ikh prilozh., Tezisy dokl. XIII Saratovskoi zimnei shkoly, Izd-vo Gos. UNTs “Kolledzh”, Saratov, 2006, 162–163

[10] Khromova G. V., “Ob otsenke pogreshnosti metoda regulyarizatsii Tikhonova dlya integralnykh uravnenii s yadrom Grina”, Vestn. MGU. Ser. 15. Vychisl. matem. i kibernetika, 1992, 22–27 | MR | Zbl

[11] Gurevich A. P., Khromov A. P., “Summiruemost po Rissu razlozhenii po sobstvennym funktsiyam integralnykh operatorov”, Izv. vuzov. Matematika, 2003, no. 2(489), 24–35 | MR | Zbl

[12] Ivanov V. K., Vasin V. V., Tanana V. P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978 | MR