Tbilisi Mathematical Journal

An alternative proof of the generalized Littlewood Tauberian theorem for Cesàro summable double sequences

Gökşen Findik, İbrahim Çanak, and Ümit Totur

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we first examine the relationships between a double sequence and its arithmetic means in different senses (i. e. $(C,1,0)$, $(C,0,1)$ and $(C,1,1)$ means) in terms of slow oscillation in certain senses and investigate some properties of oscillatory behaviors of the difference sequence between the double sequence and its arithmetic means in different senses. Next, we give an alternative proof of the generalized Littlewood Tauberian theorem for Cesàro summability method as an application of the results obtained in the first part.

Article information

Tbilisi Math. J., Volume 12, Issue 1 (2019), 131-148.

Received: 21 March 2018
Accepted: 25 January 2019
First available in Project Euclid: 26 March 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 40E05: Tauberian theorems, general
Secondary: 40G05: Cesàro, Euler, Nörlund and Hausdorff methods

Tauberian conditions and theorems convergence in Pringsheim's sense slow oscillation double sequences summability $(C,1,0)$, $(C,0,1)$ and $(C,1,1)$


Findik, Gökşen; Çanak, İbrahim; Totur, Ümit. An alternative proof of the generalized Littlewood Tauberian theorem for Cesàro summable double sequences. Tbilisi Math. J. 12 (2019), no. 1, 131--148. doi:10.32513/tbilisi/1553565632. https://projecteuclid.org/euclid.tbilisi/1553565632

Export citation


  • A. J. Badiozzaman, Some Tauberian condition on Banach Spaces, Doctoral Dissertation, University of Birmingham, United Kingdom, 1986.
  • \.I. Çanak, An extended Tauberian theorem for the $(C,1)$ summability method, Appl. Math. Lett. 21 (1) (2008), 74–80.
  • \.I. Çanak, A short proof of the generalized Littlewood Tauberian theorem, Appl. Math. Lett. 23 (7) (2010), 818–820.
  • M. Dik, Tauberian theorems for sequences with moderately oscillatory control modulo, Math. Morav. 5 (2001), 57–94.
  • K. Knopp, Limitierungs-Umkehrsätze für Doppelfolgen, Math. Z. 45 (1939), 573–589.
  • E. Landau, Über einen Satz des Herrn Littlewood, Palermo Rend. 35 (1913), 265–276.
  • F. Móricz, Tauberian theorems for Cesàro summable double sequences, Studia Math. 110 (1) (1994), 83–96.
  • F. Móricz, Ordinary convergence follows from statistical summability $(C,1)$ in the case of slowly decreasing or oscillating sequences, Colloq. Math. 99 (2) (2004), 207–219.
  • A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (3) (1900), 289–321.
  • R. Schmidt, Über divergente Folgen und lineare Mittelbildungen, Math. Z. 22 (1925), 89–152.
  • Č. V. Stanojević, Analysis of divergence: Applications to the Tauberian theory, Graduate Research Seminar, University of Missouri-Rolla, 1999.
  • O. Szász, Verallgemeinerung eines Littlewoodeschen Satzes über Potenzreihen, J. London Math. Soc. 3 (1928), 254–262.
  • A. Tauber, Ein Satz aus der Theorie der unendlichen Reihen, Monatsh. f. Math. 8 (1897), 273–277.
  • Ü. Totur, Classical Tauberian theorems for the $(C,1,1)$ summability method, An. Ştiint. Univ. Al. I. Cuza Iaşi. Mat. (N.S.). 61 (2) (2015), 401–414.
  • Ü. Totur, On the limit inferior and limit superior for double sequences, J. Class. Anal. 8 (1) (2016), 53–58.
  • T. Vijayaraghavan, A Tauberian theorem, J. Lond. Math. Soc. 1 (1926), 113–120.