An alternative proof of a Tauberian theorem for Abel summability method

Authors

  • Ibrahim Çanak Ege University.
  • Ümit Totur Adnan Menderes University.

DOI:

https://doi.org/10.4067/S0716-09172016000300001

Keywords:

Abel summability, slowly decreasing sequences, Tauberian conditions and theorems, sumabilidad abeliana, secuencias lentamente decrecientes, condiciones y teoremas de Tauber

Abstract

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

Author Biographies

Ibrahim Çanak, Ege University.

Department of Mathematics.

Ümit Totur, Adnan Menderes University.

Department of Mathematics.

References

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

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

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[8] F. Móricz, 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).

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

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

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

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

Published

2017-03-23

How to Cite

[1]
I. Çanak and Ümit Totur, “An alternative proof of a Tauberian theorem for Abel summability method”, Proyecciones (Antofagasta, On line), vol. 35, no. 3, pp. 235-244, Mar. 2017.

Issue

Section

Artículos