Uniqueness of series in the Franklin system and the Gevorkyan problems

Opuscula Mathematica

article

Zygmunt Wronicz¹ 

[1] AGH University of Science and Technology, Faculty of Applied Mathematics

关键词: Franklin system;    orthonormal spline system;    uniqueness of series.;   

DOI  :  10.7494/OpMath.2021.41.2.269

学科分类：环境科学（综合）

来源: AGH University of Science and Technology Press

【 摘 要 】

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.

【 授权许可】

CC BY-NC