Bernstein-Type Theorems and Uniqueness Theorems

• Main Authors: Logvinenko, V., Nazarova, N.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2004
• Series: Український математичний журнал
• Subjects: Статті
• Online Access: http://dspace.nbuv.gov.ua/handle/123456789/163542

Let f be an entire function of finite type with respect to finite order ρ in Cⁿ and let E be a subset of an open cone in a certain n-dimensional subspace R²ⁿ ( = Cⁿ) (the smaller ρ , the sparser E ). We assume that this cone contains a ray {z = tz⁰ ∈ Cn: t > 0} . It is shown that the radial indicator hf(z⁰) of f at any point z⁰ ∈ Cⁿ∖{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