Browsing by Subject "Franklin system"

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Access status: Open Access ,
On a problem of Gevorkyan for the Franklin system
(2016) Wronicz, Zygmunt
In 1870 G. Cantor proved that if $\lim_{N\rightarrow\infty}\sum_{n=-N}^N\,c_{n}e^{inx} = 0$ for every real $x$, where $\bar{c}_{n}=c_{n}$ $n\in \mathbb{Z}$, then all coefficients $c$ are equal to zero. Later, in 1950 V.Ya. Kozlov proved that there exists a trigonometric series for which a subsequence of its partial sums converges to zero, where not all coefficients of the series are zero. In 2004 G. Gevorkyan raised the issue that if Cantor's result extends to the Franklin system. The conjecture remains open until now. In the present paper we show however that Kozlov's version remains true for Franklin's system.
Access status: Open Access ,
Uniqueness of series in the Franklin system and the Gevorkyan problems
(Wydawnictwa AGH, 2021) Wronicz, Zygmunt
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.