Geodesic
Equiconvergence of expansions in eigenfunctions of integral operators with kernels that can have discontinuities on the~diagonals
Sbornik. Mathematics, Tome 192 (2001) no. 10, pp. 1451-1469

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Simple conditions are found ensuring the equiconvergence of the Fourier expansion of a function $f(x)$ in $L[0,1]$ in the eigenfunctions and the associated functions of an integral operator $$ Af=\int_0^{1-x}A(1-x,t)f(t)\,dt+\alpha\int_0^xA(x,t)f(t)\,dt $$ and the expansions of $f(x)$ and $f(1-x)$ in the standard trigonometric system. 
@article{SM_2001_192_10_a2,
     author = {V. V. Kornev and A. P. Khromov},
     title = {Equiconvergence of expansions in eigenfunctions of integral operators with kernels that can have discontinuities on the~diagonals},
     journal = {Sbornik. Mathematics},
     pages = {1451--1469},
     publisher = {mathdoc},
     volume = {192},
     number = {10},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2001_192_10_a2/}
}
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V. V. Kornev; A. P. Khromov. Equiconvergence of expansions in eigenfunctions of integral operators with kernels that can have discontinuities on the~diagonals. Sbornik. Mathematics, Tome 192 (2001) no. 10, pp. 1451-1469. http://geodesic.mathdoc.fr/item/SM_2001_192_10_a2/