On a problem of Gevorkyan for the Franklin system

• Autorzy: Wronicz Z.

Identyfikatory

DOI

10.7494/OpMath.2016.36.5.681

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Franklin system; orthonormal spline system; trigonometric system; uniqueness of series

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2016
• Tom: Vol. 36, no. 5
• Strony: 681--687
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

Wronicz Z.

• AGH University of Science and Technology Faculty of Applied Mathematics al. A. Mickiewicza 30, 30-059 Krakow, 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] F.G. Arutyunyan, A.A. Talalyan, Uniqueness of Series in Haar and Walsh systems, Izv. Akad. Nauk SSSR, Ser. Matem. 28 (1964), 1391-1408 [in Russian].
• [3] N.K. Bari, Trigonometric series, Moscow 1961 [in Russian].
• [4] S.V. Bochkarev, Existence of a basis in the space of functions analytic in a disc and some properties of the Franklin system, Mat. Sb. 95 (1974) 137, 3-18 [in Russian].
• [5] G. Cantor, Uber einen die Trigonometrischen Reihen betreffenden Lehrsatz, Crelles J. fur Math. 72 (1870), 130-138.
• [6] Z. Ciesielski, Properties of the orthonormal Franklin system, Studia Math. 23 (1963), 141-157.
• [7] Z. Ciesielski, A construction of basis in C(1)(/2), Studia Math. 33 (1969), 289-323.
• [8] Z. Ciesielski, Constructive function theory and spline systems, Studia Math. 53 (1975), 278-302.
• [9] Z. Ciesielski, J. Domsta, Construction of an orthonormal basis in Cm(Id) and W™(Id), Studia Math. 41 (1972), 211-224.
• [10] S. Demko, Inverses of band matrices and local convergence of splines projections, SIAM J. Numer. Anal. 14 (1977) 4, 616-619.
• [11] Ph. Franklin, A set of continuous orthogonal functions, Math. Ann. 100 (1928), 522-529.
• [12] G.G. Gevorkyan, Ciesielski and Franklin systems, [in:] Approximation and Probability, Banach Center Publ. 72, Warszawa 2006, 85-92.
• [13] V.Ya. Kozlov, On complete systems of orthogonal functions, Mat. Sbornik N.S. 26 (1950) 68, 351-364 [in Russian].
• [14] J. Marcinkiewicz, Quelques theorems sur les series orthogonales, Ann. Soc. Polon. Math. 16 (1937), 84-96.
• [15] S. Schonefeld, Schauder bases in spaces of differential functions, Bull. Amer. Math. Soc. 75 (1969), 586-590.
• [16] P. Wojtaszczyk, The Franklin system, is an unconditional basis in H1, Ark. Mat. 20 (1982), 293-300.
• [17] Z. Wronicz, On the application of the orthonormal Franklin system to the approximation of analytic functions, [in:] Approximation Theory, Banach Center Publ. 4, PWN, Warszawa 1979, 305-316.
• [18] Z. Wronicz Approximation by complex splines, Zeszyty Nauk. Uniw. Jagiellon. Prace Mat. 20 (1979), 67-88.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.