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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

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General Tauberian theorems for the Cesàro integrability of functions

Ümit Totur / İbrahim ÇanakORCID iD: https://orcid.org/0000-0002-1754-1685
Published Online: 2019-02-15 | DOI: https://doi.org/10.1515/gmj-2019-2005


For a locally integrable function f on [0,), we define


for t>0. The improper integral 0f(u)du is said to be (C,α) integrable to L for some α>-1 if the limit limxσα(t)=L exists. It is known that limtF(t)= implies limtσα(t)= for α>-1, but the converse of this implication is not true in general. In this paper, we introduce the concept of the general control modulo of non-integer order for functions and obtain some Tauberian conditions in terms of this concept for the (C,α) integrability method in order that the converse implication hold true. Our results extend the main theorems in [Ü. Totur and İ. Çanak, Tauberian conditions for the (C,α) integrability of functions, Positivity 21 2017, 1, 73–83].

Keywords: Cesàro integrability; one-sided Tauberian condition; slow decreasing; Tauberian theorem and condition

MSC 2010: 40A10; 40AC10; 40D05


  • [1]

    İ. Çanak and Ü. Totur, A Tauberian theorem for Cesàro summability of integrals, Appl. Math. Lett. 24 (2011), no. 3, 391–395. CrossrefGoogle Scholar

  • [2]

    İ. Çanak and Ü. Totur, Tauberian conditions for Cesàro summability of integrals, Appl. Math. Lett. 24 (2011), no. 6, 891–896. CrossrefGoogle Scholar

  • [3]

    İ. Çanak and Ü. Totur, Alternative proofs of some classical type Tauberian theorems for the Cesàro summability of integrals, Math. Comput. Modelling 55 (2012), no. 3–4, 1558–1561. CrossrefGoogle Scholar

  • [4]

    İ. Çanak and Ü. Totur, The (C,α) integrability of functions by weighted mean methods, Filomat 26 (2012), no. 6, 1209–1214. Web of ScienceGoogle Scholar

  • [5]

    M. Dik, Tauberian theorems for sequences with moderately oscillatory control modulo, Math. Morav. 5 (2001), 57–94. Google Scholar

  • [6]

    Y. Erdem and İ. Çanak, A Tauberian theorem for (A)(C,α) summability, Comput. Math. Appl. 60 (2010), no. 11, 2920–2925. Web of ScienceGoogle Scholar

  • [7]

    Y. Erdem and İ. Çanak, A Tauberian theorem for the product of Abel and Cesàro summability methods, Georgian Math. J. 23 (2016), no. 3, 343–350. Google Scholar

  • [8]

    R. Estrada and R. P. Kanwal, A distributional Approach to Asymptotics. Theory and Applications, 2nd ed., Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser, Boston, 2002. Google Scholar

  • [9]

    R. Estrada and J. Vindas, On Tauber’s second Tauberian theorem, Tohoku Math. J. (2) 64 (2012), no. 4, 539–560. CrossrefWeb of ScienceGoogle Scholar

  • [10]

    G. H. Hardy, Divergent Series, 2nd ed., Chelsea, New York, 1991. Google Scholar

  • [11]

    J. Korevaar, Tauberian Theory. A Century of Developments, Grundlehren Math. Wiss. 329, Springer, Berlin, 2004. Google Scholar

  • [12]

    F. Móricz and Z. Németh, Tauberian conditions under which convergence of integrals follows from summability (C,1) over 𝐑+, Anal. Math. 26 (2000), no. 1, 53–61. Google Scholar

  • [13]

    R. Schmidt, Über divergente Folgen und lineare Mittelbildungen, Math. Z. 22 (1925), no. 1, 89–152. CrossrefGoogle Scholar

  • [14]

    A. Tauber, Ein Satz aus der Theorie der unendlichen Reihen, Monatsh. Math. Phys. 8 (1897), no. 1, 273–277. CrossrefGoogle Scholar

  • [15]

    Ü. Totur and İ. Çanak, On Tauberian conditions for (C,1) summability of integrals, Rev. Un. Mat. Argentina 54 (2013), no. 2, 59–65. Google Scholar

  • [16]

    Ü. Totur and İ. Çanak, On the (C,1) summability method of improper integrals, Appl. Math. Comput. 219 (2013), no. 24, 11065–11070. Web of ScienceGoogle Scholar

  • [17]

    Ü. Totur and İ. Çanak, Tauberian conditions for the (C,α) integrability of functions, Positivity 21 (2017), no. 1, 73–83. Web of ScienceGoogle Scholar

About the article

Received: 2016-08-11

Revised: 2017-05-22

Accepted: 2017-06-28

Published Online: 2019-02-15

Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2019-2005.

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