Geodesic
Multidimensional Tauberian Theorems for Vector-valued Distributions
Publications de l'Institut Mathématique, _N_S_95 (2014) no. 109, p. 1 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of $f$ is given by the integral transform $M^f_{\varphi}(x,y)=(f*\varphi_y)(x)$, $(x,y)\in\mathbb{R}^n\times\mathbb{R}_+$, with kernel $\varphi_{y}(t)=y^{-n}ǎrphi(t/y)$. We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on $\{x_0\}\times\mathbb R^m$. In addition, we present a new proof of Littlewood's Tauberian theorem.
Classification : 40E05 41A27 26A12 40E10 41A60 42C40 46F10 46F12
Keywords: Abelian and Tauberian theorems, vector-valued distributions, quasiasymptotics, slowly varying functions, Laplace transform, wavelet transform, regularizing transforms, asymptotic behavior of generalized functions 
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     author = {Stevan Pilipovi\'c and Jasson Vindas},
     title = {Multidimensional {Tauberian} {Theorems} for {Vector-valued} {Distributions}},
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Stevan Pilipović; Jasson Vindas. Multidimensional Tauberian Theorems for Vector-valued Distributions. Publications de l'Institut Mathématique, _N_S_95 (2014) no. 109, p. 1 . http://geodesic.mathdoc.fr/item/PIM_2014_N_S_95_109_a0/