Ukrainian Mathematical Journal

, Volume 56, Issue 2, pp 244–263 | Cite as

Bernstein-Type Theorems and Uniqueness Theorems

  • V. Logvinenko
  • N. Nazarova


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.


Entire Function Approximation Technique Uniqueness Theorem Finite Type Finite Order 
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Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • V. Logvinenko
    • 1
  • N. Nazarova
    • 2
  1. 1.Pasadena City CollegePasadenaUSA
  2. 2.Kharkov Polytechnic UniversityKharkov

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