TY  - JOUR
ID  - Wronicz2021
LB  - Wronicz2021
AU  - Wronicz, Zygmunt
TI  - Uniqueness of series in the Franklin system and the Gevorkyan problems
ST  - Uniqueness of series in the Franklin system and the Gevorkyan problems
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2021
VL  - 41
IS  - 2
SP  - 269
EP  - 276
DO  - https://doi.org/10.7494/OpMath.2021.41.2.269
UR  - https://doi.org/10.7494/OpMath.2021.41.2.269
AB  - 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.
KW  - Franklin system
KW  - orthonormal spline system
KW  - uniqueness of series
SN  - 1232-9274
ER  -