On completeness of exponential systems in convex domain

The work is devoted to studying the completeness of the systems of exponentials in the space of functions analytic in a convex domain. The problem on the completeness of the systems of the exponentials in various functional spaces is classical and was studied by many mathematicians, for instance, by B.Ya. Levin, A.F. Leontiev, A.M. Sedletskii, B.N. Khabibullin, R.S. Yulmukhametov, and others.
We prove that the completeness of the system of exponentials in the space of functions analytic in a convex domain is equivalent to the completeness of the system of exponentials in the space of functions analytic in a circle with the radius depending on the properties of a given convex domain. We also consider an example by choosing an ellipse as the convex domain. Here we find the values of the support function and the radius of the corresponding circle.