Geodesic
Inversion of integral operators with kernels discontinuous on the diagonal
Matematičeskie zametki, Tome 64 (1998) no. 6, pp. 932-942

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Conditions implying the invertibility of the integral operator $$ Af(x)=\int_0^1A(x,t)f(t)\,dt $$ with kernel $A(x,t)$ having discontinuities of the first kind at the points $t=x$ and $t=1-x$ are found. We give explicit inversion formulas as well as applications to the problem of finding the square roots of the operator $y''(x)$ with arbitrary boundary conditions and the problem of expansion with respect to eigenfunctions. 
@article{MZM_1998_64_6_a13,
     author = {A. P. Khromov},
     title = {Inversion of integral operators with kernels discontinuous on the diagonal},
     journal = {Matemati\v{c}eskie zametki},
     pages = {932--942},
     publisher = {mathdoc},
     volume = {64},
     number = {6},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_6_a13/}
}
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A. P. Khromov. Inversion of integral operators with kernels discontinuous on the diagonal. Matematičeskie zametki, Tome 64 (1998) no. 6, pp. 932-942. http://geodesic.mathdoc.fr/item/MZM_1998_64_6_a13/