Uniqueness Theorem for Entire Functions of Exponential Type

Issues of completeness of various functional systems in spaces of analytical functions constitute an important and actively developing area of complex analysis. According to the general principles of duality theory, the problem often reduces to the study of uniqueness sets in classes of entire functions with growth restrictions. Thus, in a recent joint work by Braichev, Khabibullin, and Sherstyukov, three different approaches to constructing uniqueness sets for classes of entire functions with restrictions on their indicator and type were proposed. One approach is related to Sylvester’s famous problem of the smallest circle containing a given set of points in the plane and theorems of convex geometry. Namely, in the mentioned paper the following result was proved: the sequence of zeros of an entire function of exponential type of completely regular growth forms a uniqueness set for the class of entire functions whose exponential type is less than the radius of the Sylvester circle constructed for the indicator diagram of the generative function. We weaken the condition for complete regularity of growth in this theorem. As it turned out, we can limit ourselves to the requirement that the generative function for the sequence under consideration has a completely regular growth only on two or three specially chosen rays. To do this, the concept of a collection of Sylvester rays associated with the greatest growth of a function is introduced. To identify examples that support the theory, Azarin’s profound results on the existence of an entire function with given upper and lower growth indicatrices are used.