Geodesic
An Equiconvergence Theorem for an Integral Operator with a~Variable Upper Limit of Integration
Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 182-191.

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We suggest simple sufficient conditions on the kernel of the integral operator $$ Af=\int\limits_0^{1-x}A(1-x,t)f(t)\,dt $$ providing expansion with respect to the root functions to be equiconvergent with ordinary Fourier series. 
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A. P. Khromov. An Equiconvergence Theorem for an Integral Operator with a~Variable Upper Limit of Integration. Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 182-191. http://geodesic.mathdoc.fr/item/CMFD_2007_25_a14/

[1] Khromov A. P., “Teoremy ravnoskhodimosti dlya integro-differentsialnykh i integralnykh operatorov”, Matem. sb., 114(156):3 (1981), 378–405 | MR | Zbl

[2] Khromov A. P., “Ob obraschenii integralnykh operatorov s yadrami, razryvnymi na diagonalyakh”, Sovremennye problemy teorii funktsii i ikh prilozheniya, Tezisy dokladov 9-i Saratovskoi zimnei shkoly, Saratov, 1997, 162