An inhomogeneous system of convolution equations in complex domains (Q2455225)

Denote by \(U\) the vertical strip \(\{z: |\Re z|<1\},\) by \(H(U)\) the space of holomorphic functions on this strip with the topology of uniform convergence on compact sets, and by \(H^{\ast}(U)\) its strong dual. Suppose that functionals \(S_1, S_2, \dots , S_k \in H^{\ast}(U)\) are such that the entire functions of exponential type \(\varphi_j(\lambda)=\langle S_j, e^{\lambda z}\rangle\), \(j=1,2, \dots, k,\) satisfy the relations \[ \limsup _{r \to \pm \infty}\frac {\ln|\varphi_j(r)|}{|r|}=d_j, \quad d_j\geq 0; \;d_j+d_i<1, \;i,j=1,2, \dots,k, \;i\neq j. \] Let \(U_j\) be the vertical strip \(\{z: |\Re z|<1-d_j\}\), \(j=1,2,\ldots,k.\) Let \(\varphi_1\) be a function of completely regular growth and its zeros \(\{\lambda_n\}\) are real, simple, and separated, i.e., there exists a number \(c>0\) such that \(|\lambda_n-\lambda_m|\geq c\) for \(n\neq m.\) In the paper the authors obtain a criteria of solvability of the system of convolution equations \[ (S_j\ast f)(z)=\langle S_j,f(z+h)\rangle=g_j(z), \quad j=1,2,\dots, k, \] where the functions \(g_j \in H(U_j)\) are given, and it is necessary to find the function \(f\in H(U).\) To formulate results of this paper we need some notation and notions. Let \(\{\mu_n\}\) be the zeros of the function \(\varphi_1\) that are not zeros of \(|\varphi_2(\lambda)|+ \dots +|\varphi_k(\lambda)|,\) and let \(\{\nu_n\}\) be the remaining zeros. Definition. The right-hand sides of the above system are admissible, if they satisfy the following conditions: \[ S_i\ast g_j=S_j\ast g_i, \;(i,j)=1,2,\ldots,k, \;i\neq j;\quad T_1 \ast g_j=T_j \ast g_1, \] where \(T_1\) and \(T_j\) are linear continuous functionals whose Laplace transforms are equal, respectively, \[ \frac{\varphi_1(\lambda)}{\lambda-\nu_m} \quad \text{and}\quad \frac{\varphi_j(\lambda)}{\lambda-\nu_m}, \;j=2,3,\ldots,k, \;m\in \mathbb{N}. \] (These necessary solvability conditions were given by A. S. Krivosheev.) The authors prove the following Theorem. The above system is solvable for any admissible right-hand side from the space \(\prod_{j=1}^{k}H(U_j)\) if and only if \[ \lim_{n \to \infty} \max_{j=2,3,\dots,k} \left(\frac{\ln|\varphi_j(\mu_n)|}{|\mu_n|}-d_j\right)=0. \] Analogous criteria for the solvability of this system in a half-plane and in the whole plane are also obtained.