2013/04/15 by Stevan Pilipovic, Jasson Vindas
Mathematics · #math.FA #math.CA #msc:40E05 #msc:41A27. #msc:26A12 #msc:40E10 #msc:41A60 #msc:42C40 #msc:46F10 #msc:46F12
paper · pdf · doi:10.2298/pim1409001p
published as Publ. Inst. Math. (Beograd) 95 (2014), 1-28 · 28 pages. arXiv admin note: substantial text overlap with arXiv:1012.5090
arxiv created 2013/04/15 · arxiv updated 2014/07/25
We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of f is given by the integral transform Mfφ(x,y)=(f∗φy)(x), (x,y)∈ℝn×ℝ+, with kernel φy(t)=y-nφ(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 \x0\× \mathbb Rm. In addition, we present a new proof of Littlewood's Tauberian theorem.