The structure of quasiasymptotics of Schwartz distributions

In this article complete characterizations of quasiasymptotic behaviors 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 the quasiasymptotic of degree -1 and it is shown how the structural theorem can be used to study Ces\`{a}ro and Abel summability of trigonometric series and integrals. Further properties of quasiasymptotics at infinity are discussed, the author presents a condition over test functions which allows one to evaluate them at the quasiasymptotic, these test functions are in bigger spaces than $\mathcal{S}$. 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.