Geodesic
On Convergence of Expansions in Eigen Functions of Integral Operators with Discontinuous Kernel
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 1, pp. 59-62.

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For integral operators with a jump of its kernel on the diagonal it will be found necessary and sufficient conditions of invertibility. Conditions providing equiconvergence of expansions in eigen functions of these operators and trigonometric Fourier series are established. 
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V. V. Kornev. On Convergence of Expansions in Eigen Functions of Integral Operators with Discontinuous Kernel. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 1, pp. 59-62. http://geodesic.mathdoc.fr/item/ISU_2013_13_1_a14/

[1] Khromov A. P., “Integral operators with kernels that are discontinuous on broken lines”, Sb. Math., 197:11 (2006), 1669–1696 | DOI | DOI | MR | Zbl

[2] Hromov A. P., “Equiconvergence theorems for integrodifferential and integral operators”, Math. USSR Sb., 42:3 (1982), 331–355 | DOI | MR