Wiener-Type Tauberian Theorems for Fourier Hyperfunctions

Abstract

Two Wiener-type Tauberian theorems concerning Fourier hyperfunctions are proved and commented. It is shownt that the shift asymptotics ($S$-asymptotics) of a hyperfunction $f$ is determined by the ordinary asymptotics of $(f\star \mathcal{K})(x)$ as $x\to \infty$, where $\mathcal{K}$ is Hörmaner’s kernel. Moreover, Wiener-type theorems are used for the asympthotic analysis of solutions to some (pseudo-)differential equations.