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Abstract

In this article complete characterizations of the quasiasymptotic behavior of Schwartz distributions are presented by means of structural theorems. The cases at infinity and the origin are both analyzed. Special attention is paid to quasiasymptotics of degree $-1$. It is shown how the structural theorem can be used to study Cesàro and Abel summability of trigonometric series and integrals. Further properties of quasiasymptotics at infinity are discussed. A condition for test functions in bigger spaces than $\mathcal{S}$ is presented which allows one to consider the respective quasiasymptotics over them. An extension of the structural theorems for quasiasymptotics is given. The author studies a structural characterization of the behavior $f(\lambda x)=O(\rho(\lambda))$ in $\mathcal{D'}$, where $\rho$ is a regularly varying function.