SciELO - Scientific Electronic Library Online

 
vol.35 issue3Unicyclic graphs with equal domination and complementary tree domination numbers author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.35 no.3 Antofagasta Sept. 2016

http://dx.doi.org/10.4067/S0716-09172016000300001 

 

An alternative proof of a Tauberian theorem for Abel summability method

 

Ibrahim Çanak

Ege University, Turkey

Ümit Totur

Adnan Menderes University, Turkey 


ABSTRACT

Using a corollary to Karamata’s main theorem [Math. Z. 32 (1930), 319-320], we prove that ifa slowly decreasing sequence ofreal numbers is Abel summable, then it is convergent in the ordinary sense.

Subjclass [2010] : 40A05; 40E05; ]0G10.

Keywords : Abel summability, slowly decreasing sequences, Tauberian conditions and theorems.


REFERENCES

[1]    G. H. Hardy, Divergent series, Oxford University Press, (1948).         [ Links ]

[2]    J. Karamata, Uber die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z., 32, pp. 319-320, (1930).         [ Links ]

[3]    K. Knopp, Theory and application of infinite series, Dover Publications, (1990).         [ Links ]

[4]    J. Korevaar, Tauberian theory, Springer, 2004.         [ Links ]

[5]    J. E. Littlewood, The converse of Abel’s theorem on power series, London M. S. Proc. 2 (9), pp. 434-448, (1911).

[6]    I. J. Maddox, A Tauberian theorem for ordered spaces, Analysis, 9, (3), pp. 297-302, (1989).         [ Links ]

[7]    G. A. Mikhalin, Theorem of Tauberian type for (J,pn) summation methods, Ukrain. Mat. Zh. 29 (1977), 763-770. English translation: Ukrain. Math. J. 29, pp. 564-569, (1977).         [ Links ]

[8]    F. Moricz, Ordinary convergence follows from statistical summability (C, 1) in the case of slowly decreasing or oscillating sequences, Colloq. Math. 99, (2), pp. 207-219, (2004).         [ Links ]

[9]    R. Schmidt, Uber divergente Folgen und lineare Mittelbildungen, Math. Z. 22 (1), pp. 89-152, (1925).         [ Links ]

[10]    C. V. Stanojevic, V. B. Stanojevic, Tauberian retrieval theory, Publ. Inst. Math. (Beograd) (N.S.) 71 (85), pp. 105-111, (2002).         [ Links ]

[11]    O. Talo, F. Basar, On the slowly decreasing sequences of fuzzy num-bers, Abstr. Appl. Anal. Art. ID 891986, 7, pp. ..., (2013).         [ Links ]

[12]    A. Tauber, Ein satz aus der theorie der unendlichen reihen, Monatsh. f. Math. u. Phys. 7, pp. 273-277, (1897).         [ Links ]


Received : December 2014. Accepted : July 2016

 

Ibrahim Çanak
Department of Mathematics
Ege University
Izmir
Turkey
e-mail : ibrahim.canak@ege.edu.tr

Ümit Totur
Department of Mathematics
Adnan Menderes University
 Aydin
 Turkey

e-mail : utotur@adu.edu.tr

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License