## Tohoku Mathematical Journal 2012 December SECOND SERIES VOL. 64, NO. 4

 Tohoku Math. J. 64 (2012), 539-560

Title ON TAUBER'S SECOND TAUBERIAN THEOREM

Author Ricardo Estrada and Jasson Vindas

(Received October 14, 2010, revised November 28, 2011)
Abstract. 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.

2000 Mathematics Subject Classification. Primary 40E05; Secondary 40G05, 40G10, 46F10, 46F20.

Key words and phrases. 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.