| Popis: |
A classic theorem of Pólya shows that 2z is, in a strong sense, the "smallest" transcendental entire function that is integer valued on ℕ. An analogous result of Gel'fond concerns entire functions that are integer valued on the set Xa = {an: n ∈ ℕ}, where a ∈ ℕ, |a| ≥ 2. Let X = ℕ or X = Xa and k ∈ ℕ or k = ∞. This paper pursues analogous results for entire functions f having the following property: on any finite subset D of X with #D ≤ k + 1, the values f(z),z ∈ D admit interpolation by an element of ℤ[z]. The results obtained assert that if the growth of f is suitably restricted then the restriction of f to X must be a polynomial. When X = Xa and k < ∞ a "smallest" transcendental entire function having the requisite property is constructed. |
