Wiener Tauberian theorems for ultradistributions
type: Item
instance of: scholarly article
title: Wiener Tauberian theorems for ultradistributions
zbMATH Open document ID: 0827.46033
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1995-12-17T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108337
review text: Summary: The purpose of this paper is the extension of Wiener Tauberian theorems for distributions on ultradistribution spaces. Because of that, we give the versions of Beurling's and Wiener's theorems for bounded ultradistributions. The corollary of our main theorem is the following one. Let \(f\) be an ultradistribution such that \(f/c\) is a bounded ultradistribution, where \(c\) is a smooth function which behaves as \(L(e^x) e^{\alpha x}\), \(x\to \infty\), \(L\) is a slowly varying function at \(\infty\) and \(\alpha\in \mathbb{R}\). If for an ultradifferentiable function \(\varphi\) with the property \({\mathcal F} [\varphi ](\xi- i\alpha) \neq 0\), \(\xi\in \mathbb{R}\), \[ \lim_{x\to\infty} {{(f* \varphi) (x)} \over {L(e^x) e^{\alpha x}}}= a\int \varphi (t) e^{-\alpha t} dt, \qquad a\in \mathbb{R}, \] then for every ultradifferentiable function \(\psi\) \[ {{(f* \psi) (x)} \over {L(e^x) e^{\alpha x}}}\to a\int \psi (t) e^{- \alpha t} dt, \qquad x\to \infty. \]
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 46F12
zbMATH DE Number: 763302
zbMATH Keywords: Wiener Tauberian theorems
zbMATH Keywords: ultradistribution spaces
author: Stevan Pilipović
author: B. Stanković
MaRDI profile type: Publication
cites work: Q3992980
cites work: Q5581948
cites work: Q5673937
cites work: Q4288766
cites work: Characterizations of Bounded Sets in Spaces of Ultradistributions
cites work: Wiener Tauberian Theorems for Distributions
cites work: Q3843128
description: scientific article; zbMATH DE number 763302