Geodesic
Absolute convergence of expansions in~eigen- and adjoint functions of
Izvestiya. Mathematics , Tome 69 (2005) no. 4, pp. 703-717

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We establish an analogue of Zygmund's criterion for the absolute convergence of trigonometric Fourier series for expansions in eigen- and adjoint functions of the integral operator $Af(x)=\int_0^{1-x}A(1-x,t)f(t)\,dt$. 
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     author = {V. V. Kornev and A. P. Khromov},
     title = {Absolute convergence of expansions in~eigen- and adjoint functions of},
     journal = {Izvestiya. Mathematics },
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V. V. Kornev; A. P. Khromov. Absolute convergence of expansions in~eigen- and adjoint functions of. Izvestiya. Mathematics , Tome 69 (2005) no. 4, pp. 703-717. http://geodesic.mathdoc.fr/item/IM2_2005_69_4_a2/