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On Absolute Summability by Riesz and Generalized Cesàro Means. I

Published online by Cambridge University Press:  20 November 2018

H.-H. Körle
H.-H. Körle*
Affiliation:
Universitât Marburg Marburg, Germany
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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 → ∞.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 202 - 208
Copyright
Copyright © Canadian Mathematical Society 1970

References

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