Geodesic
A Tauberian Theorem for the General Euler-Borel Summability Method
Canadian journal of mathematics, Tome 44 (1992) no. 5, pp. 1100-1120

Voir la notice de l'article provenant de la source Cambridge University Press

Our main result is a Tauberian theorem for the general Euler-Borel summability method. Examples of the method include the discrete Borel, Euler, Meyer- Kônig, Taylor and Karamata methods. Every function/ analytic on the closed unit disk and satisfying some general conditions generates such a method, denoted by (£,ƒ). For instance the function ƒ(z) = exp(z — 1) generates the discrete Borel method. To each such function ƒ corresponds an even positive integer p = p(f).We show that if a sequence (sn)is summable (E,f)and as n→ ∞ m > n, (m— n)n-p(f) →0, then (sn)is convergent. If the Maclaurin coefficients of/ are nonnegative, then p(f) =2. In this case we may replace the condition . This generalizes the Tauberian theorems for Borel summability due to Hardy and Littlewood, and R. Schmidt. We have also found new examples of the method and proved that the exponent —p(f)in the Tauberian condition (*) is the best possible.
DOI : 10.4153/CJM-1992-067-9
Mots-clés : Primary: 40E05, 40G05, 40G10 
Tam, Laying. A Tauberian Theorem for the General Euler-Borel Summability Method. Canadian journal of mathematics, Tome 44 (1992) no. 5, pp. 1100-1120. doi: 10.4153/CJM-1992-067-9
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