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On Absolute Summability by Riesz and Generalized Cesàro Means. I

  • H.-H. Körle (a1)

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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.

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References

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1. Borwein, D., On a generalised Cesàro summability method of integral order, Töhoku Math. J. 18 (1966), 7173.
2. Borwein, D., On a method of summability equivalent to the Cesàro method, J. London Math. Soc. 42 (1967), 339343.
3. Borwein, D. and Russell, D. C., On Riesz and generalised Cesàro summability of arbitrary positive order, Math. Z. 99 (1967), 171177.
4. Burkill, H., On Riesz and Riemann summability, Proc. Cambridge Philos. Soc. 57 (1961), 5560.
5. Chandrasekharan, K. and Minakshisundaram, S., Typical means (Oxford Univ. Press, London, 1952).
6. Hardy, G. H., Divergent series (Oxford Univ. Press, London, 1956/1963).
7. Hardy, G. H. and Riesz, M., The general theory of Dirichlet's series, Cambridge Tract no. 18 (University Press, Cambridge, 1915/1952).
8. Jurkat, W., Uber Rieszsche Mittel und verwandte Klassen von Matrixtransformationen, Math. Z. 57 (1953), 353394.
9. Kogbetliantz, E., Sommation des séries et intégrales divergentes par les moyennes arithmétiques et typiques, Mémorial Sci. Math. 51 (Gauthier-Villars, Paris, 1931).
10. Körle, H.-H., On the equivalence of Riesz and generalized Cesàro ﹛absolute) summability. II, Notices Amer. Math. Soc. 15 (1968), 923.
11. Meir, A., An inclusion theorem for generalized Cesàro and Riesz means, Can. J. Math. 20 (1968), 735738.
12. Obrechkoff, N., Uber die absolute Summierung der Dirichletschen Reihen, Math. Z. 30 (1929), 375386.
13. Russell, D. C., On generalized Cesàro means of integral order, Töhoku Math. J. 17 (1965), 410-442; Corrigenda, ibid. 18 (1966), 454455.
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On Absolute Summability by Riesz and Generalized Cesàro Means. I

  • H.-H. Körle (a1)

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