Geodesic
An analog of the Jordan--Dirichlet theorem for an integral operator whose kernel has discontinuites on the diagonals
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 20 International Saratov Winter School "Contemporary Problems of Function Theory and Their Applications", Saratov, January 28 — February 1, 2020. Part 2, Tome 200 (2021), pp. 87-94.

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In the paper, we examine an integral operator whose kernel has first-kind discontinuites at the lines t=x and t=1−x. For this operator, we prove an analog of the Jordan–Dirichlet theorem on the convergence of eigenfunction expansion. The convergence is studied using the method based on integration of the resolvent by the spectral parameter.
Mots-clés : Jordan–Dirichlet theorem, resolvent, eigenfunction. 
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E. V. Nazarova; V. A. Khalova. An analog of the Jordan--Dirichlet theorem for an integral operator whose kernel has discontinuites on the diagonals. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 20 International Saratov Winter School "Contemporary Problems of Function Theory and Their Applications", Saratov, January 28 — February 1, 2020. Part 2, Tome 200 (2021), pp. 87-94. https://geodesic-test.mathdoc.fr/item/INTO_2021_200_a9/

[1] Bari N. K., Trigonometricheskie ryady, Fizmatgiz, M., 1961

[2] Burlutskaya M. Sh., “Analog teoremy Zhordana—Dirikhle dlya operatora differentsirovaniya na grafe”, Vestn. Voronezh. un-ta. Ser. Fiz. Mat., 2006, no. 2, 165–168 | Zbl

[3] Golub A. V., “Analog teoremy Zhordana—Dirikhle dlya integralnogo operatora obschego vida s involyutsiei, imeyuschei razryv”, Mat. 17 Mezhdunar. Saratov. zimnei shkoly «Sovremennye problemy teorii funktsii i ikh prilozheniya» (Saratov, 27 yanvarya — 3 fevralya 2014 g.), Nauchnaya kniga, Saratov, 2014, 78–79

[4] Gurevich A. P., Khromov A. P., “Summiruemost po Rissu spektralnykh razlozhenii odnogo klassa integralnykh operatorov”, Differ. uravn., 37:6 (2001), 809–814 | MR | Zbl

[5] Kornev V. V., Khromov A. P., “O ravnoskhodimosti razlozhenii po sobstvenym funktsiyam integralnykh operatorov s yadrami, dopuskayuschimi razryvy proizvodnykh na diagonalyakh”, Mat. sb., 192:10 (2001), 33–50 | Zbl

[6] Koroleva O. A., “Teorema Zhordana—Dirikhle dlya operatora differentsirovaniya”, Matematika. Mekhanika., 2013, no. 15, 40–43

[7] Koroleva O. A., “Analog teoremy Zhordana—Dirikhle dlya integralnogo operatora s yadrom, imeyuschim skachki na lomanykh liniyakh”, Izv. Saratov. un-ta. Nov. ser. Ser. Mat. Mekh. Inform., 3:4-1 (2013), 14–22

[8] Nazarova E. V., “Zadacha tochnogo obrascheniya nekotorogo klassa integralnykh operatorov s razryvnymi yadrami”, Mat. 10 Mezhdunar. Saratov. zimnei shkoly «Sovremennye problemy teorii funktsii i ikh prilozheniya», Izd-vo Saratov. un-ta, Saratov, 2000, 96–97

[9] Nazarova E. V., “O ravnoskhodimosti razlozhenii po sobstvennym funktsiyam integralnykh operatorov s yadrami, razryvnymi po diagonalyakh”, Matematika. Mekhanika., 2002, no. 4, 102–105

[10] Nazarova E. V., Khalova V. A., “Teorema ravnoskhodimosti dlya integralnogo operatora s involyutsiei”, Izv. Saratov. un-ta. Nov. ser. Ser. Mat. Mekh. Inform., 17:3 (2017), 313–330 | MR | Zbl

[11] Khalova V. A., “Ob analoge teoremy Zhordana—Dirikhle dlya razlozhenii po sobstvennym funktsiyam odnogo klassa differentsialno-raznostnykh operatorov”, Izv. Saratov. un-ta. Nov. ser. Ser. Mat. Mekh. Inform., 10:3 (2010), 26–32

[12] Khromov A. P., “Smeshannaya zadacha dlya volnovogo uravneniya s proizvolnymi dvukhtochechnymi kraevymi usloviyami”, Dokl. RAN., 462:2 (2015), 148–150 | MR | Zbl

[13] Khromov A. P., Lukomskii S. F., Sidorov S. P., Terekhin P. A., Novye metody approksimatsii v zadachakh deistvitelnogo analiza i spektralnoi teorii, Izd-vo Saratov. un-ta, Saratov, 2015

[14] Khromov A. P., “On an analog of the Jordan–Dirichlet theorem for eigenfunction expansions of one differential-difference operator with an integral boundary condition”, J. Math. Sci., 144:4 (2007), 4267–4276 | DOI | MR | Zbl