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Some Tauberian theorems for (A)\((C,\alpha )\) summability method. (English) Zbl 1202.40007

Summary: We give some Tauberian conditions in order to obtain \((C,\alpha )\) summability from \((A)(C,\alpha )\) summability.

MSC:

40G05 Cesàro, Euler, Nörlund and Hausdorff methods
40E05 Tauberian theorems
40G10 Abel, Borel and power series methods
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References:

[1] Hardy, G. H., Divergent Series (1991), Chelsea: Chelsea New York, NY · Zbl 0897.01044
[2] Abel, N. H., Recherches sur la série \(1 + \frac{m}{1} x + \frac{m(m - 1)}{1.2} x^2 + \cdots \), J. für Math., 1, 311-339 (1826) · ERAM 001.0031cj
[3] Tauber, A., Ein satz der Theorie der unendlichen Reihen, Monatsh. f. Math., 8, 273-277 (1897) · JFM 28.0221.02
[4] Littlewood, J. E., The converse of Abel’s theorem on power series, Proc. Lond. Math. Soc., 9, 2, 434-448 (1911) · JFM 42.0276.01
[5] Hardy, G. H.; Littlewood, J. E., Tauberian theorems concerning power and Dirichlet’s series whose coefficients are positive, Proc. Lond. Math. Soc., 13, 2, 174-191 (1913) · JFM 45.0389.02
[6] Szàsz, O., Generalization of two theorems of Hardy and Littlewood on power series, Duke Math. J., 1, 105-111 (1935) · JFM 61.0217.03
[7] Dik, M., Tauberian theorems for sequences with moderately oscillatory control modulo, Math. Morav., 5, 57-94 (2001) · Zbl 1046.40004
[8] Pati, T., On tauberian theorems, (Rath, D.; Nanda, S., Sequences, Summability and Fourier Analysis (2005), Narosa Publishing House), 84-96
[9] Kogbetliantz, E., Sur le séries absolument sommables par la méthode des moyennes arihtmétiques, Bull. Soc. Math., 49, 2, 234-251 (1925) · JFM 51.0182.01
[10] Kogbetliantz, E., Sommation des séries et intégrals divergents par les moyennes arithmétiques et typiques, Memorial Sci. Math., 51, 1-84 (1931) · JFM 57.1376.02
[11] Hardy, G. H., Theorems relating to the summability and convergence of slowly oscillating series, Proc. Lond. Math. Soc., 8, 2, 301-320 (1910) · JFM 41.0278.02
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