Geodesic
Distributional versions of Littlewood's Tauberian theorem
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 2, pp. 403-420.

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We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We employ two types of Tauberian hypotheses; the first kind involves distributional boundedness, while the second type imposes a one-sided assumption on the Cesàro behavior of the distribution. We apply these Tauberian results to deduce a number of Tauberian theorems for power series and Stieltjes integrals where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.
DOI : 10.1007/s10587-013-0025-1
Classification : 40E05, 40G05, 40G10, 44A10, 46F12, 46F20
Keywords: Tauberian theorem; Laplace transform; the converse of Abel's theorem; Littlewood's Tauberian theorem; Abel and Cesàro summability; distributional Tauberian theorem; asymptotic behavior of generalized function 
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     title = {Distributional versions of {Littlewood's} {Tauberian} theorem},
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Estrada, Ricardo; Vindas, Jasson. Distributional versions of Littlewood's Tauberian theorem. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 2, pp. 403-420. doi : 10.1007/s10587-013-0025-1. http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0025-1/

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