Geodesic
Multidimensional Tauberian Theorems for Vector-valued Distributions
Publications de l'Institut Mathématique, _N_S_95 (2014) no. 109, p. 1 Cet article a éte moissonné depuis la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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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}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {1 },
     year = {2014},
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     number = {109},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_2014_N_S_95_109_a0/}
}
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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/