Factorization of a~convolution-type operator

V. V. Napalkov

Matematičeskie zametki, Tome 15 (1974) no. 1, pp. 165-171.

Résumé

Three convolution-type equations are considered in the space of entire functions with topology ofd uniform convergence: \begin{gather*} M_{\mu_1}[f]=\int_Cf(z+t)d\mu_1=0\\ M_{\mu_2}[f]=\int_Cf(z+t)d\mu_2=0\\ M_\mu[f]=\int_Cf(z+t)d\mu=0 \end{gather*} with respective characteristic functions $L_1(\lambda)$, $L_2(\lambda)$, $L(\lambda)=L_1(\lambda)\cdot L_2(\lambda)$, $\operatorname{supp}\mu\Subset C$, $\operatorname{supp}\mu_1\Subset C$, $\operatorname{supp}\mu_2\Subset C$. The necessary and sufficient conditions are found that every solution $f(z)$ of the equation $M_\mu[f]=0$ can be written as a sum $f_1(z)+f_2(z)$, where $f_1(z)$ is the solution of the equation $M_{\mu_1}[f]=0$, $f_2(z)$ is the solution of the equation $M_{\mu_2}[f]=0$.