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

Zygmunt Wronicz

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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1. J.H. Ahlberg, E.N. Nilson, J.L. Walsh, The Theory of Splines and Their Applications, Academic Press, New York - London, 1967.
2. G. Cantor, Über einen die Trigonometrischen Reihen betreffenden Lehrsatz, Crelles J. für Math. 72 (1870), 130-138.
3. Z. Ciesielski, Properties of the orthonormal Franklin system, Studia Math. 23 (1963), 141-157.
4. Ph. Franklin, A set of continuous orthogonal functions, Math. Ann. 100 (1928), 522-529.
5. G.G. Gevorkyan, Ciesielski and Franklin systems, [in:] T. Figiel, A. Kamont (eds.), Approximation and Probability, Banach Center Publ. 72 (2006), 85-92.
6. G.G. Gevorkyan, Uniqueness theorems for series in the Franklin system, Mat. Zametki 98 (2015), 786-789 [in Russian], English transl. in Math. Notes 98 (2015), 847-851.
7. G.G. Gevorkyan, Uniqueness theorems for series in the Franklin system, Sbornik: Math. 207 (2016), 1650-1673 [in Russian], English transl. in Math. Sb. 207 (2016), 30-53.
8. F. Leja, Rachunek różniczkowy i całkowy, PWN, Warszawa, 1959 [in Polish].
9. Z. Wronicz, On a problem of Gevorkyan for the Franklin system, Opuscula Math. 36 (2016), 681-687.
10. Z. Wronicz, On the application of the orthonormal Franklin system to the approximation of analytic functions, [in:] Z. Ciesielski (ed.), Approximation Theory 4 (1979), 305-316.
11. Z. Wronicz, Approximation by complex splines, Zeszyty Nauk. Uniw. Jagiellon., Prace Mat. 20 (1979), 67-88.
• Communicated by Zdzisław Jackiewicz.
• Revised: 2021-01-24.
• Accepted: 2021-01-25.
• Published online: 2021-03-17.