Geodesic
Convergence of orthogonal series to $+\infty$
Matematičeskie zametki, Tome 8 (1970) no. 2, pp. 129-136

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For any sequence $\{b_n\}$ such that $\sum_{n=1}^\infty b_n^2=\infty$, a uniformly bounded system $\{\Phi_n(x)\}$, orthonormal on $[0, 1]$, is constructed such that the series $\sum_{n=1}^\infty b_n\Phi_n(x)$ diverges to $+\infty$ on some set $E\subset[0, 1]$, $0\mathrm{mes}\, E1$, for any order of the terms. 
@article{MZM_1970_8_2_a0,
     author = {R. I. Ovsepyan and A. A. Talalyan},
     title = {Convergence of orthogonal series to $+\infty$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {129--136},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_2_a0/}
}
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R. I. Ovsepyan; A. A. Talalyan. Convergence of orthogonal series to $+\infty$. Matematičeskie zametki, Tome 8 (1970) no. 2, pp. 129-136. http://geodesic.mathdoc.fr/item/MZM_1970_8_2_a0/