An alternative proof of a Tauberian theorem for Abel summability method

  • Ibrahim Çanak Ege University.
  • Ümit Totur Adnan Menderes University.
Palabras clave: Abel summability, slowly decreasing sequences, Tauberian conditions and theorems, sumabilidad abeliana, secuencias lentamente decrecientes, condiciones y teoremas de Tauber

Resumen

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.

Biografía del autor

Ibrahim Çanak, Ege University.
Department of Mathematics.
Ümit Totur, Adnan Menderes University.
Department of Mathematics.

Citas

[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).

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

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

[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).

[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).

[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).
Publicado
2017-03-23
Cómo citar
Çanak, I., & Totur, Ümit. (2017). An alternative proof of a Tauberian theorem for Abel summability method. Proyecciones. Journal of Mathematics, 35(3), 235-244. https://doi.org/10.4067/S0716-09172016000300001
Sección
Artículos