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An Inclusion Relation for Abel, Borel, and Lambert Summability

Published online by Cambridge University Press:  20 November 2018

W. Gawronski ,
H. Siebert  and
R. Trautner
W. Gawronski
Affiliation:
Universität Ulm, Oberer Eselsberg, Germany
H. Siebert
Affiliation:
Universität Ulm, Oberer Eselsberg, Germany
R. Trautner
Affiliation:
Universität Ulm, Oberer Eselsberg, Germany
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In this paper a new type of inclusion theorem concerning Abel, Borel and Lambert summability is established. To state our results we need some definitions and notations. With a formal series ∑k=0∞ak, ak∈ C, and its partial sums snwe associate the series

Then ∑k=0∞ ak is said to be summable to the value s

(a) by Abel's method, if (1.1) is convergent for |v| > 1 and limv→1+A(v)= s,

(b) by Lambert's method, if (1.2) is convergent for |v| > 1 and limv→1+L(v)= s,

(c) by Borel's method, if (1.3) is convergent for all x ∈ R and limx→+∞B(x)= s,

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 3 , 01 June 1980 , pp. 695 - 702
Copyright
Copyright © Canadian Mathematical Society 1980

References

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