A Characterization for Fourier Hyperfunctions

Abstract

The space of test functions for Fourier hyperfunctions is characterized by two conditions $\sup |\phi (x)|\exp k|x|<\infty$ and $\sup |\hat{\phi }(\xi )|\exp h|\xi |<\infty$ for some $h,k>0$. Combining this result and the new characterization of Schwartz space in [1] we can easily compare two important spaces $\mathcal{F}$ and $\mathcal{S}$ which are both invariant under Fourier transformations.