Opuscula Math. 36, no. 5 (2016), 681-687
http://dx.doi.org/10.7494/OpMath.2016.36.5.681

Opuscula Mathematica

# On a problem of Gevorkyan for the Franklin system

Zygmunt Wronicz

Abstract. 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 $$c_{n}$$ 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.

Keywords: Franklin system, orthonormal spline system, trigonometric system, uniqueness of series.

Mathematics Subject Classification: 42C10, 42C25, 41A15.

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• Zygmunt Wronicz
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland
• Communicated by P.A. Cojuhari.
• Revised: 2016-01-09.
• Accepted: 2016-05-15.
• Published online: 2016-06-29.