Fourier Hyperfunctions as the Boundary Values of Smooth Solutions of Heat Equations

Abstract

We show that if a $C^{\infty }$-solution $u(x,t)$ of heat equation in ${\mathbf{R}}_{+}^{n+1}$ does not increase faster than $\exp [\epsilon (\frac{1}{t}+|x|)]$ then its boundary value determines a unique Fourier hyperfunction. Also, we prove the decomposition theorem for the Fourier hyper functions. These results generalize the theorems of T. Kawai and T. Matsuzawa for Fourier hyperfunctions and solve a question given by A. Kaneko.