Article
MSC:
40E05,
40G05,
40G10,
44A10,
46F12,
46F20 | MR 3073966 | Zbl 06236419 | DOI: 10.1007/s10587-013-0025-1
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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
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
Summary:
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.
References:
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