On approximations of the solutions of a homogeneous convolution equation with several unknown functions (Q1902713)

Let \({\mathfrak G} = \{G_1, \ldots, G_q\}\) be a system of \(q\) convex domains \(G_i \subset \mathbb{C}\). We relate to this system a locally convex space \(H = H_1 \times \cdots \times H_q\), where \(H_i\) is a space of holomorphic in \(G_i\) functions allotted by customary topology of uniform convergence on compacts. In what follows, \(H^*_i\) stands for the strong conjugated space of \(H_i\). We consider the homogeneous convolution equation \[ S_1*f_1 + \cdots + S_q*f_q = 0, \tag{1} \] where \(f_i \in H_i\), \(S_i \in H^*_i\), \(i = 1, \ldots, q\). The convolution of a functional \(S_i\) and a function \(f_i\) is defined by the rule \((S_i*f_i)(h) = \langle S_i,f_i(z + h) \rangle\). It is a function holomorphic in some neighborhood of the origin. A solution to equation (1) is a \(q\)-vector \(f = (f_1, \ldots, f_q) \in H\) whose components satisfy the equation. A solution to equation (1) is said to be elementary if it has the form \(a_0 e^{\lambda z} + a_1 ze^{\lambda z} + \cdots + a_n z^ne^{\lambda z}\), where \(a_0, a_1, \ldots, a_n\) are \(q\)- dimensional vectors with complex components, and the multiplication by functions \(e^{\lambda z}\), \(ze^{\lambda z}, \ldots, z^ne^{\lambda z}\) is component-wise. It is known that any solution \(f \in H\) to equation (1) can be approximated by a linear combination of elementary solutions in topology. The present article deals with the following question. Let the components \(f_1, \ldots, f_q\) of a solution \(f \in H\) have one-valued analytic continuation into simply connected domains \(G_1', \ldots, G_q'\) \((G_i \subset G_i'\), \(i = 1, \ldots, q)\), respectively. We find conditions for the approximability of \(f\) by combinations of elementary solutions in the topology of the space \(H'\) associated with the system of domains \({\mathfrak G}' = \{G_1', \ldots, G_q'\}\). Below we describe all the systems \({\mathfrak G}'\) of simply connected domains for which such an approximation is possible under assumption of complete regularity of the growth of the components \(\varphi_i (h) = \langle S_i, e^{hz} \rangle\) of the characteristic element \(\varphi = (\varphi_1, \ldots, \varphi_q)\) of equation (1) along rays from certain sets. These systems coincide with systems of ``flag'' domains introduced by the author in Mat. Zametki 32, 199-211 (1982; Zbl 0514.45012).