A characterization for Fourier hyperfunctions (Q1331245)

Summary: The space of test functions for Fourier hyperfunctions is characterized by two conditions \(\sup|\varphi(x)| \exp k| x|< \infty\) and \(\sup|\widehat\varphi(\xi)| \exp h|\xi|< \infty\) for some \(h, k>0\). Combining this result and the new characterization of Schwartz space in [the authors, C. R. Acad. Sci., Paris, Ser. I 316, No. 1, 23-25 (1993)] we can easily compare two important spaces \(\mathcal F\) and \(\mathcal S\) which are both invariant under Fourier transformations.