Wiener Tauberian theorems for ultradistributions
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 \(\psi\) \[ {{(f* \psi) (x)} \over {L(e^x) e^{\alpha x}}}\to a\int \psi (t) e^{- \alpha t} dt, \qquad x\to \infty. \]
scientific article; zbMATH DE number 763302