Multipliers for Laplace hyperfunctions – a justification of Heaviside's rules

Holomorphic functions of exponential type on an open sector containing $[a,\infty]$ are multipliers for Laplace hyperfunctions with support in $[a,\infty]$. Their action in the Laplace images is realized as convolutions in the complex domain. In the special case of the exponential $e^{\omega x}$ and the coordinate $x$ it is the shift by $\omega$ and the derivation $-d/d\lambda$ respectively. Bessel's equation is treated as an example.