Uniqueness of series in the Franklin system and the Gevorkyan problems

• Autorzy: Wronicz Zygmunt

Identyfikatory

DOI

10.7494/OpMath.2021.41.2.269

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Franklin system; orthonormal spline system; uniqueness of series

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2021
• Tom: Vol. 41, no. 2
• Strony: 269--276
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Wronicz Zygmunt

• AGH University of Science and Technology Faculty of Applied Mathematics al. A. Mickiewicza 30, 30-059 Kraków, Poland

Bibliografia

• [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, Uber einen die Trigonometrischen Reihen betreffenden Lehrsatz, Crelles J. fur 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 Mat. 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, Banach Center Publ. 4 (1979), 305-316.
• [11] Z. Wronicz, Approximation by complex splines, Zeszyty Nauk. Uniw. Jagiellon., Prace Mat. 20 (1979), 67-88.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).