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Wiener Tauberian theorems for ultradistributions. (English) Zbl 0827.46033
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 $$\psi$$ ${{(f* \psi) (x)} \over {L(e^x) e^{\alpha x}}}\to a\int \psi (t) e^{- \alpha t} dt, \qquad x\to \infty.$

##### MSC:
 46F05 Topological linear spaces of test functions, distributions and ultradistributions 46F12 Integral transforms in distribution spaces