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Ukrainian Mathematical Journal

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

Bernstein-Type Theorems and Uniqueness Theorems

  • V. Logvinenko
  • N. Nazarova
Article
  • 34 Downloads

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.

Keywords

Entire Function Approximation Technique Uniqueness Theorem Finite Type Finite Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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