Abstract
We characterize possible bounds for representing functions of arbitrary hyperfunctions. Specifically, there are always representing functions decreasing rapidly outside each strip near $\mathbb{R}$. Also, exponential decrease of any type on any strip $\mathbb{R}\times \pm i[c,C], 0<c<C<\infty,$ can be achieved. This will be used in [10] to define an asymptotic Fourier and Laplace transformation on the space of hyperfunctions.
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