Growth of entire functions of exponential type and characteristics of distributions of points along straight line in complex plane

Abstract

According a classical Weierstrass-Hadamard-Lindelöf theorem, for each distribution of points with a finite upper density in the complex plane, there exists a non-zero entire function of exponential type vanishing on the these points with the multiplicity taken into account. In the beginning of 1960s, in a joint work by P. Malliavin and L.A. Rubel, the following problem was completely solved. Given two distributions of points on the positive half-line with finite upper densities, find relations between these distributions under which for each non-zero entire function of exponential type vanishing on one of the distributions, there exists a non-zero entire function of exponential type vanishing on the other distribution and having the absolute value not exceeding that of the first function. A complete solution of this problem going back to works by F. Carlson, T. Carleman, M. Cartwright, L. Schwartz, J.-P. Kahane and many others, was given in terms of so-called logarithmic characteristics of distributions of points, which are expressed via reciprocals to points in these distributions. In this paper we extend these results on complex distributions of the points separated from the imaginary axis by a pair of vertical angles of an arbitrary small opening; here we develop logarithmic characteristics for complex distributions of points. We consider three types of possible restrictions on the growth along the imaginary axis, very strict ones, as by P. Malliavin and L.A. Rubel, and less restrictive as in previous works by the second co-author. The main results are of a completed form and are formulated as criterions.