Geodesic
On the order of partial sums of general orthogonal series
Matematičeskie zametki, Tome 6 (1969) no. 4, pp. 451-462 
R. S. Davtyan. On the order of partial sums of general orthogonal series. Matematičeskie zametki, Tome 6 (1969) no. 4, pp. 451-462. http://geodesic.mathdoc.fr/item/MZM_1969_6_4_a10/
@article{MZM_1969_6_4_a10,
     author = {R. S. Davtyan},
     title = {On the order of partial sums of general orthogonal series},
     journal = {Matemati\v{c}eskie zametki},
     pages = {451--462},
     year = {1969},
     volume = {6},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_6_4_a10/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is shown that for convergence of every orthonormal system $\{\varphi_n(s)\}$ given on $[0,1]$, it is necessary and sufficient that, under the condition $\int_0^\infty\frac1{W^2(x)}dx<+\infty$ on tlie increasing function $W(x)$ and for $\sum_{n=1}^\infty a_n^2=+\infty$ there hold $\left|\sum_{k=1}^na_k\varphi_k(x)\right|=o(W(\sum_1^ka_k^2))$ almost everywhere on $[0,1]$.