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Uniqueness of series in the Franklin system and the Gevorkyan problems

Wronicz Zygmunt

Opuscula Mathematica

2021

Vol. 41, no. 2

269-276

In 1870 G. Cantor proved that if [formula], then cn = 0 for n ∈ 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 [formula] it suffices to prove the convergence its subsequence s2n to zero by the condition [formula]. It is a solution of the Gevorkyan problem formulated in 2016.

On a problem of Gevorkyan for the Franklin system

Wronicz Z.

Opuscula Mathematica

2016

Vol. 36, no. 5

681--687

In 1870 G. Cantor proved that if [formula] for every real x, where [formula] then all coefficients cn 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.