Opuscula Math. 41, no. 2 (2021), 269-276
https://doi.org/10.7494/OpMath.2021.41.2.269
Opuscula Mathematica
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.
Keywords: Franklin system, orthonormal spline system, uniqueness of series.
Mathematics Subject Classification: 42C10, 42C25, 41A15.
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- Zygmunt Wronicz
https://orcid.org/0000-0003-1221-0817
- AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Kraków, Poland
- Communicated by Zdzisław Jackiewicz.
- Received: 2020-12-01.
- Revised: 2021-01-24.
- Accepted: 2021-01-25.
- Published online: 2021-03-17.