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.