Geodesic
On a uniqueness theorem for the Franklin system
Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 52 (2018) no. 2, pp. 93-100

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In this paper we prove that there exist a nontrivial Franklin series and a sequence$M_n$ such that the partial sums$S_{M_n}(x)$ of that series converge to 0 almost everywhere and $\lambda\cdot \mathrm{mes}\left\{x:sup_n\big|S_{M_n}(x)\big|>\lambda\right\}\to 0$ as $\lambda\to+\infty$. This shows that the boundedness assumption of the ratio $M_{n+1} /M_n$, used for the proofs of uniqueness theorems in earlier papers, can not be omitted.
Keywords: majorant of partial sums, Franklin system, uniqueness. 
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     author = {K. A. Navasardyan},
     title = {On a uniqueness theorem for the {Franklin} system},
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K. A. Navasardyan. On a uniqueness theorem for the Franklin system. Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 52 (2018) no. 2, pp. 93-100. http://geodesic.mathdoc.fr/item/UZERU_2018_52_2_a3/