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Tauberian class estimates for vector-valued distributions
S. Pilipovića, J. Vindasb a Department of Mathematics and Informatics, University of Novi Sad, Novi Sad, Serbia
b Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
Abstract:
We study Tauberian properties of regularizing transforms of vector-valued tempered distributions, that is, transforms of the form $M^{\mathbf{f}}_{\varphi}(x,y)=(\mathbf{f}\ast\varphi_{y})(x)$, where the kernel $\varphi$ is a test function and $\varphi_{y}(\cdot)=y^{-n}\varphi(\cdot/y)$. We investigate conditions which ensure that a distribution that a priori takes values in locally convex space actually takes values in a narrower Banach space. Our goal is to characterize spaces of Banach space valued tempered distributions in terms of so-called class estimates for the transform $M^{\mathbf{f}}_{\varphi}(x,y)$. Our results generalize and improve earlier Tauberian theorems of Drozhzhinov and Zav'yalov (Sb. Math. 194 (2003), 1599–1646). Special attention is paid to find the optimal class of kernels $\varphi$ for which these Tauberian results hold.
Keywords:
regularizing transforms, class estimates, Tauberian theorems, vector-valued distributions, generalized functions, wavelet transform.
DOI:
https://doi.org/10.4213/sm9061
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DOI: https://doi.org/10.1070/SM9061
Document Type:
Article
UDC:
517.53
MSC: Primary 40E05, 46F05; Secondary 46F12 Received: 05.01.2018
Citation:
S. Pilipović, J. Vindas, “Tauberian class estimates for vector-valued distributions”, Mat. Sb., 210:2 (2019), 115–142
Citation in format AMSBIB
\Bibitem{PilVin19}
\by S.~Pilipovi{\'c}, J.~Vindas
\paper Tauberian class estimates for vector-valued distributions
\jour Mat. Sb.
\yr 2019
\vol 210
\issue 2
\pages 115--142
\mathnet{http://mi.mathnet.ru/msb9061}
\crossref{https://doi.org/10.4213/sm9061}
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http://mi.mathnet.ru/eng/msb9061https://doi.org/10.4213/sm9061 http://mi.mathnet.ru/eng/msb/v210/i2/p115
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