A tauberian theorem for distributional point values

Rendiconti Del Seminario Matematico, 2013

We give the following version of Fatou's theorem for distributions that are boundary values of analytic functions. We prove that if $f\in\mathcal{D}^{\prime}(a,b) $ is the distributional limit of the analytic function $F$ defined in a region of the form $(a,b) \times(0,R),$ if the one sided distributional limit exists, $f(x_{0}+0) =\gamma,$ and if $f$ is distributionally bounded at $x=x_{0}$, then the \L ojasiewicz point value exists, $f(x_{0})=\gamma$ distributionally, and in particular $F(z)\to \gamma$ as $z\to x_{0}$ in a non-tangential fashion.

Tohoku Mathematical Journal, 2012

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 (C, β) summability for a given order β. 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 Szász for Dirichlet series. ∞ n=0 c n = γ (A) . Mathematics Subject Classification. Primary 40E05, 40G05, 40G10, 46F20; Secondary 46F10. 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.

Czechoslovak Mathematical Journal, 2013

We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series 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.

Tamkang Journal of Mathematics, 2004

We give a method to construct distributions that are boundary values of analytic functions which have non-tangential limits at points where the distributional point value does not exist.

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1983

We consider the determination of the behavior of a distribution function F at its endpoints in terms of the behavior of its Laplace-Stieltjes transform co at the limits of its interval of convergence. The results extend various known strong and weak results to a larger class of distributions via a relatively straightforward technique based on the weak convergence of suitably normalized associated distributions. An application and examples are considered briefly.