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A Tauberian Theorem for Borel-Type Methods of Summability
Published online by Cambridge University Press: 20 November 2018
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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.
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- Copyright © Canadian Mathematical Society 1969
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