Tohoku Mathematical Journal

On Tauber's second Tauberian theorem

Ricardo Estrada and Jasson Vindas

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We study Tauberian conditions for the existence of Cesàro limits in terms of the Laplace transform. We also analyze Tauberian theorems for the existence of distributional point values in terms of analytic representations. The development of these theorems is parallel to Tauber's second theorem on the converse of Abel's theorem. For Schwartz distributions, we obtain extensions of many classical Tauberians for Cesàro and Abel summability of functions and measures. We give general Tauberian conditions in order to guarantee $(\mathrm{C},\beta)$ summability for a given order $\beta$. The results are directly applicable to series and Stieltjes integrals, and we therefore recover the classical cases and provide new Tauberians for the converse of Abel's theorem where the conclusion is Cesàro summability rather than convergence. We also apply our results to give new quick proofs of some theorems of Hardy-Littlewood and Szàsz for Dirichlet series.

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Tohoku Math. J. (2), Volume 64, Number 4 (2012), 539-560.

First available in Project Euclid: 20 December 2012

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Zentralblatt MATH identifier

Primary: 40E05: Tauberian theorems, general
Secondary: 40G05: Cesàro, Euler, Nörlund and Hausdorff methods 40G10: Abel, Borel and power series methods 46F10: Operations with distributions 46F20: Distributions and ultradistributions as boundary values of analytic functions [See also 30D40, 30E25, 32A40]

Tauberian theorems the converse of Abel's theorem Hardy-Littlewood Tauberians Szász Tauberians distributional point values boundary behavior of analytic functions asymptotic behavior of generalized functions Laplace transform Cesàro summability


Estrada, Ricardo; Vindas, Jasson. On Tauber's second Tauberian theorem. Tohoku Math. J. (2) 64 (2012), no. 4, 539--560. doi:10.2748/tmj/1356038977.

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