Bernstein-Type Theorems and Uniqueness Theorems
Let \(f\) be an entire function of finite type with respect to finite order \(ρ in C^n \) and let \(E\) be a subset of an open cone in a certain n-dimensional subspace \(R^2n ( = C^n ) \) (the smaller \(ρ \) , the sparser \(E\) ). We assume that this cone contains a ray \({ z = tz^0 C^n :t > 0 }\) . It is shown that the radial indicator \(h_f (z^0 )\) of \(f\) at any point \(z^0 C^n { 0} \) may be evaluated in terms of function values at points of the discrete subset \(E\) . Moreover, if \(f\) tends to zero fast enough as \(z → ∞ \) over \(E\) , 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 \(ρ \) and \(E\) , which are close to exact conditions, the function \(f\) bounded on \(E\) is bounded on the ray., number=2, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Logvinenko, V. and Nazarova, N., year=2004, month=Feb., pages=198–213