We present a criterion ensuring that solutions \(\varphi \in {\mathscr {C}}^\infty (\mathbb {R}^{n+1}_{t,{\varvec{x}}},\mathbb {C})\) of a system of convolution equations \(\mu _0 * \varphi = \cdots = \mu _M *\varphi =0\) in \(\mathbb {R}^{n+1}\) , where the \(\mu _j\) ’s are distributions with compact support whose Fourier transforms satisfy \({\text {codim}}_{\mathbb {C}^{n+1}} \{\widehat{\mu }_j=0\,:\,j=0,\ldots ,M\}=M+1\) , can be represented in any Euclidean ball \(\mathbb {B}_{\mathbb {R}^{n+1}}(0,R)\) as the superposition of (almost) elementary solutions. Our approach is based on Bochner–Martinelli weighted integral formulas when applied to Lagrange interpolation or division. In this setting, weighted integral formulas combine with the Atiyah realization of principal value or residue Paley–Wiener tempered distributions. Such a criterion is motivated by the example where the \(\mu _j\) ’s are difference-delayed operators with delays restricted to the t variable. The analytic criterion we propose relies on slicing, thus challenging the geometric criterion (dictated by the Cauchy–Weil integral representation formula) introduced by C. A. Berenstein and B. A. Taylor as the slowly decreasing constraint. Bibliography : 21 titles.
