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**On approximations of the solutions of a homogeneous convolution equation with several unknown functions.**
*(English.
Russian original)*
Zbl 0830.45011

Russ. Math. 37, No. 1, 19-24 (1993); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1993, No. 1, 21-26 (1993).

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).

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).

### MSC:

45N05 | Abstract integral equations, integral equations in abstract spaces |

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

45L05 | Theoretical approximation of solutions to integral equations |