Geodesic
On a Property of the Franklin System in~$C[0,1]$ and $L^1[0,1]$
Matematičeskie zametki, Tome 107 (2020) no. 2, pp. 241-245

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A problem posed by J. R. Holub is solved. In particular, it is proved that if $\{\widetilde f_n\}$ is the normalized Franklin system in $L^1[0,1]$, $\{a_n\}$ is a monotone sequence converging to zero, and $\sup_{n\in\mathbb N}\|{\sum_{k=0}^na_k\widetilde f_k}\|_1+\infty$, then the series $\sum_{n=0}^{\infty}a_n\widetilde f_n$ converges in $L^1[0,1]$. A similar result is also obtained for $C[0,1]$.
Keywords: Franklin system, bounded completeness, monotonically bounded completeness. 
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     author = {V. G. Mikayelyan},
     title = {On a {Property} of the {Franklin} {System} in~$C[0,1]$ and $L^1[0,1]$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {241--245},
     publisher = {mathdoc},
     volume = {107},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_107_2_a6/}
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V. G. Mikayelyan. On a Property of the Franklin System in~$C[0,1]$ and $L^1[0,1]$. Matematičeskie zametki, Tome 107 (2020) no. 2, pp. 241-245. http://geodesic.mathdoc.fr/item/MZM_2020_107_2_a6/