Geodesic
On the convergence of orthogonal series to $+\infty$
Matematičeskie zametki, Tome 11 (1972) no. 5, pp. 499-508.

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For any sequence of numbers $a_n\downarrow0$, $\sum_{n=1}^\infty a_n^2=\infty$, a uniformly bounded orthonormal system of continuous functions $\varphi_n(x)$ which is complete in $L_2(0,1)$, and a sequence of numbers $b_n$ ($0$) are constructed such that $\sum_{n=1}^\infty b_n\varphi_n(x)=\infty$ everywhere on $(0, 1)$. 
@article{MZM_1972_11_5_a3,
     author = {R. I. Ovsepyan},
     title = {On the convergence of orthogonal series to $+\infty$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {499--508},
     publisher = {mathdoc},
     volume = {11},
     number = {5},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_11_5_a3/}
}
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R. I. Ovsepyan. On the convergence of orthogonal series to $+\infty$. Matematičeskie zametki, Tome 11 (1972) no. 5, pp. 499-508. http://geodesic.mathdoc.fr/item/MZM_1972_11_5_a3/