Regular growth of systems of functions and systems of non-homogeneous convolution equations in convex domains of the complex plane (Q2710715)

Let \(G\subset\mathbb C\) be a convex domain and let \(\mu_1,\dots,\mu_n\) be analytic functionals in \(\mathbb C\). Let \(f_j\) be the Laplace transform of \(\mu_j\) (\(f_j(z):=(\mu_j)_y(\exp(yz))\), \(z\in\mathbb C\)) and let \(K_j\) be the conjugate diagram of \(f_j\), \(j=1,\dots,n\). Put \(D:=G+K_1+\dots+K_n\), \(G_j:=G+K_1+\dots+K_{j-1}+K_{j+1}+\dots+K_n\). The author characterizes the existence of a solution \(\psi\in\mathcal O(D)\) of the system of non-homogeneous convolution equations \((\mu_j)_y(\psi(y+z))=g_j(z)\), \(z\in G_j\), \(j=1,\dots,n\) (\(g_j\in\mathcal O(G_j)\), \(j=1,\dots,n\), are given) in terms of the growth of the functions \(f_1,\dots,f_n\).