On a Tauberian theorem for the weighted mean method of summability
Keywords:
Summability by weighted means, Tauberian conditionsAbstract
We investigate conditions needed for a weighted mean summable series to beconvergent by using Kloosterman’s method. The results of this paper generalize thewell known results of Landau and Hardy.References
Çanak, İ. & Totur, Ü. 2011. Some Tauberian theorems for the weighted mean methods of summability,
Computers and Mathematics with Applications, 62(6):2609-2615.
Çanak, İ. & Totur, Ü. 2013. Extended Tauberian theorem for the weighted mean method of summability,
Ukrainian Mathematical Journal, 65(7):1032-1041.
Dik, M. 2001. Tauberian theorems for sequences with moderately oscillatory control modulo, Mathematica
Moravica, 5:57-94.
Hardy, G. H. 1910. Theorems relating to the summability and convergence of slowly oscillating series,
Proceedings of the London Mathematical Society (2), 8:301-320.
Kloosterman, H. D. 1940. On the convergence of series summable (C, r) and on the magnitude of the
derivatives of a function of a real variable, Journal of the London Mathematical Society, 15:91-96.
Landau, E. 1910. Über die Bedeutung einiger neuen Grenzwertsätze der Herren Hardy und Axer., Prace
Matematyczno-Fizyczne,21:97–177.
Móricz, F. & Rhoades B. E. 1995. Necessary and sufficient Tauberian conditions for certain weighted
mean methods of summability, Acta Mathematica Hungarica, 66(1-2):105-111.
Móricz, F. & Rhoades B. E. 2004. Necessary and sufficient Tauberian conditions for certain weighted
mean methods of summability II., Acta Mathematica Hungarica, 102(4):279–285.
Móricz, F. & Stadtmüller U. 2001. Necessary and sufficient conditions under which convergence follows
from summability by weighted means, International Journal of Mathematics and Mathematical
Sciences, 27(7): 399–406.
Rau, K. A. 1930. An example in the theory of summation of series by Riesz’s typical means, Proceedings
of the London Mathematical Society (2), 30:367–372.
Tietz, H. 1990. Schmidtsche Umkehrbedingungen für Potenzreihenverfahren, Acta Scientiarum
Mathematicarum, 54(3-4): 355–365.
Tietz, H. & Zeller K. 1998. Tauber-Bedingungen für Verfahren mit Abschnittskonvergenz, Acta
Mathematica Hungarica, 81(3): 241–247.
Totur, Ü. & Çanak, İ. 2012. Some general Tauberian conditions for the weighted mean summability
method, Computers and Mathematics with Applications, 63(5): 999-1006.
