Locally explicit version of Fundamental Principle for homogeneous convolution equations

Using a weighted Koppelman integral representation formula with a complex parameter a division formula in the space of weighted entire function is proved. This formula is used to derive a locally explicit version of Ehrenpreis's Fundamental Principle for a system of homogeneous convolution equations $\check{f}\ast \mu_j=0$, $j=1,\dots, m$, $f \in \mathcal{E}(\mathbb R^n)$, $\mu_j\in\mathcal{E}^{\prime}(\mathbb R^n)$, when the Fourier Transforms $\hat{\mu}_j$, $j=1,\dots, m$ are slowly decreasing and form a complete intersection in $\mathbb C^n$.