An alternative proof of a Tauberian theorem for Abel summability method
DOI:
https://doi.org/10.4067/S0716-09172016000300001Keywords:
Abel summability, slowly decreasing sequences, Tauberian conditions and theorems, sumabilidad abeliana, secuencias lentamente decrecientes, condiciones y teoremas de TauberAbstract
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.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).
[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).
[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).
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.
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