Indicators of products and surjective convolution maps (Q793182)

Let f(z) be an entire function of exponential type with indicator h(t,f) and conjugate diagram \(P_ f\). Denote by H(G) the space of functions analytic in the convex region G endowed with the topology of uniform convergence on compact subsets. Let \(\hat f:H(G+P_ f)\to H(G)\) be a convolution-type operator generated by the function f(z). The main results of the paper are the following: Theorem 1. Fix the function f(z) and \(t\in [0,2\pi]\). For any entire function g(z) of exponential type the equality \(h(t,fg)=h(t,f)+h(t,g)\) holds, if and only if f(z) is of completely regular growth on the ray \(\arg z=t.\) Theorem 2. Fix the function f(z). For any convex region G the convolution-type operator \(\hat f\) is an epimorphism if and only if f(z) is of completely regular growth. Reviewer's remark. Both theorems are known in a more general form [cf. \textit{V. S. Azarin}, Teor. Funkts., Funkts. Anal. Prilozh. 2, 55-68 (1966; Zbl 0241.30033) and \textit{V. A. Tkachenko}, Izv. Akad. Nauk SSSR, Ser. Mat. 41, 378-392 (1977; Zbl 0356.45006)]. The second paper contains the references to the papers by V. V. Napalkov and O. V. Epifanov. The former proved Theorem 2 independently and the latter gave its generalization for a fixed region G.