## 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).

### 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

Zbl 0514.45012