Uniqueness of series in the Franklin system and the Gevorkyan problems

Abstract

In 1870 G. Cantor proved that if $\lim_{N \rightarrow \infty}\sum_{n=-N}^N c_{n}e^{inx} = 0$, $\bar{c}_{n}=c_{n}$, then $c_{n}=0$ for $n \in \mathbb{Z}$. In 2004 G. Gevorkyan raised the issue that if Cantor's result extends to the Franklin system. He solved this conjecture in 2015. In 2014 Z. Wronicz proved that there exists a Franklin series for which a subsequence of its partial sums converges to zero, where not all coefficients of the series are zero. In the present paper we show that to the uniqueness of the Franklin system $\lim_{n\rightarrow \infty}\sum_{n=0}^\infty a_{n}f_{n}$ it suffices to prove the convergence its subsequence $s_{2^{n}}$ to zero by the condition $a_{n}=o(\sqrt{n})$. It is a solution of the Gevorkyan problem formulated in 2016.