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A Tauberian Theorem for Borel-Type Methods of Summability

Published online by Cambridge University Press:  20 November 2018

D. Borwein
D. Borwein*
Affiliation:
University of Western Ontario, London, Ontario
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Suppose throughout that α >0, β is real, and Nis a non-negative integer such that αN+ β> 0. A series of complex terms is said to be summable (B, α,β) to l if, as x→ ∞,

where sn= a0 + a1 + … + an.The Borel-type summability method (B, α, β) is regular, i.e., all convergent series are summable (B, α,β) to their natural sums; and (B,1, 1) is the standard Borel exponential method B.

Our aim in this paper is to prove the following Tauberian theorem.

THEOREM. Iƒ

(i) p ≧ – ½, an = o(np), and

(ii) is summable (B, α,β) to l, then the series is summable by the Cesaro method(C, 2p + 1) to l.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 740 - 747
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Borwein, D., On methods of summability based on integral Junctions. II, Proc. Cambridge Philos. Soc. 56 (1960), 125–131 Google Scholar
2. Borwein, D., Relations between Borel-type methods of summability, J. London Math. Soc. 35 (1960), 65–70.Google Scholar
3. Hardy, G. H., Divergent series (Oxford, at the Clarendon Press, 1949).Google Scholar
4. Whittaker, E. T. and Watson, G. N., A course of modern analysis, 4th ed. (Cambridge Univ. Press, Cambridge, 1927).Google Scholar