Bernstein-Type Theorems and Uniqueness Theorems

Abstract

Let \(f\) be an entire function of finite type with respect to finite order \(\rho {\text{ in }}\mathbb{C}^n \) and let \(\mathbb{E}\) be a subset of an open cone in a certain n-dimensional subspace \(\mathbb{R}^{2n} {\text{ ( = }}\mathbb{C}^n {\text{)}}\) (the smaller \(\rho \) , the sparser \(\mathbb{E}\) ). We assume that this cone contains a ray \(\left\{ {z = tz^0 \in \mathbb{C}^n :t > 0} \right\}\) . It is shown that the radial indicator \(h_f (z^0 )\) of \(f\) at any point \(z^0 \in \mathbb{C}^n \backslash \{ 0\} \) may be evaluated in terms of function values at points of the discrete subset \(\mathbb{E}\) . Moreover, if \(f\) tends to zero fast enough as \(z \to \infty \) over \(\mathbb{E}\) , then this function vanishes identically. To prove these results, a special approximation technique is developed. In the last part of the paper, it is proved that, under certain conditions on \(\rho \) and \(\mathbb{E}\) , which are close to exact conditions, the function \(f\) bounded on \(\mathbb{E}\) is bounded on the ray.