The translation-asymptotic and the quasiasymptotic behaviour of a distribution (Q1174429)

We prove that the translation-asymptotic behaviour of an \(f\in{\mathcal S}_ +'\) with respect to \(k^ \nu L(k)\) (\(k>0\)), \(\nu\leq-1\) implies the quasiasymptotic behaviour of \(f\) at \(\infty\) of order \(-1\). This assertion shows that for \(\nu\leq-1\) the translation-asymptotic and the quasiasymptotic behaviour of a distribution are not comparable. If \(\nu>- 1\) then the translation-asymptotic behaviour of an \(f\in{\mathcal S}_ +'\) with respect to \(k^ \nu L(k)\) (\(k>0\)) implies its quasiasymptotic behaviour with respect to the same function [cf. \textit{V. S. Vladimirov}, \textit{Yu. N. Drozyzhimov} and \textit{B. I. Zav'yalov}, ``Multidimensional Tauber theorem for generalized functions'' (in Russian) (1986; Zbl 0605.40006)].