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.
Keywords: Jordan–Dirichlet theorem, resolvent, eigenfunction. 
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     author = {E. V. Nazarova and V. A. Khalova},
     title = {An analog of the {Jordan--Dirichlet} theorem for an integral operator whose kernel has discontinuites on the diagonals},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {87--94},
     publisher = {mathdoc},
     volume = {200},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_200_a9/}
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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. http://geodesic.mathdoc.fr/item/INTO_2021_200_a9/