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. 
@article{CMFD_2007_25_a14,
     author = {A. P. Khromov},
     title = {An {Equiconvergence} {Theorem} for an {Integral} {Operator} with {a~Variable} {Upper} {Limit} of {Integration}},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {182--191},
     publisher = {mathdoc},
     volume = {25},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2007_25_a14/}
}
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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/