Geodesic
On Absolute Summability by Riesz and Generalized Cesàro Means. I
Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 202-208

Voir la notice de l'article provenant de la source Cambridge University Press

1. The Cesàro methods for ordinary [9, p. 17; 6, p. 96] and for absolute [9, p. 25] summation of infinite series can be generalized by the Riesz methods [7, p. 21; 12; 9, p. 52; 6, p. 86; 5, p. 2] and by “the generalized Cesàro methods“ introduced by Burkill [4] and Borwein and Russell [3]. (Also cf. [2]; for another generalization, see [8].) These generalizations raise the question as to their equivalence.We shall consider series (1) with complex terms an. Throughout, we will assume that (2) and we call (1) Riesz summable to a sum s relative to the type λ = (λn ) and to the order κ, or summable (R, λ, κ) to s briefly, if the Riesz means (of the partial sums of (1)) tend to s as x → ∞. 
Körle, H.-H. On Absolute Summability by Riesz and Generalized Cesàro Means. I. Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 202-208. doi: 10.4153/CJM-1970-026-6
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[1] 1. Borwein, D., On a generalised Cesàro summability method of integral order, Töhoku Math. J. 18 (1966), 71–73. Google Scholar

[2] 2. Borwein, D., On a method of summability equivalent to the Cesàro method, J. London Math. Soc. 42 (1967), 339–343. Google Scholar

[3] 3. Borwein, D. and Russell, D. C., On Riesz and generalised Cesàro summability of arbitrary positive order, Math. Z. 99 (1967), 171–177. Google Scholar

[4] 4. Burkill, H., On Riesz and Riemann summability, Proc. Cambridge Philos. Soc. 57 (1961), 55–60. Google Scholar

[5] 5. Chandrasekharan, K. and Minakshisundaram, S., Typical means (Oxford Univ. Press, London, 1952). Google Scholar

[6] 6. Hardy, G. H., Divergent series (Oxford Univ. Press, London, 1956/1963). Google Scholar

[7] 7. Hardy, G. H. and Riesz, M., The general theory of Dirichlet's series, Cambridge Tract no. 18 (University Press, Cambridge, 1915/1952). Google Scholar

[8] 8. Jurkat, W., Uber Rieszsche Mittel und verwandte Klassen von Matrixtransformationen, Math. Z. 57 (1953), 353–394. Google Scholar

[9] 9. Kogbetliantz, E., Sommation des séries et intégrales divergentes par les moyennes arithmétiques et typiques, Mémorial Sci. Math. 51 (Gauthier-Villars, Paris, 1931). Google Scholar

[10] 10. Körle, H.-H., On the equivalence of Riesz and generalized Cesàro ﹛absolute) summability. II, Notices Amer. Math. Soc. 15 (1968), 923. Google Scholar

[11] 11. Meir, A., An inclusion theorem for generalized Cesàro and Riesz means, Can. J. Math. 20 (1968), 735–738. Google Scholar

[12] 12. Obrechkoff, N., Uber die absolute Summierung der Dirichletschen Reihen, Math. Z. 30 (1929), 375–386. Google Scholar

[13] 13. Russell, D. C., On generalized Cesàro means of integral order, Töhoku Math. J. 17 (1965), 410-442; Corrigenda, ibid. 18 (1966), 454–455. Google Scholar

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