On a theorem of Pólya and Levinson
Let \(f\) be an entire function in \(\mathbb{C}\). If for any \(\varepsilon>0\) there exists a positive constant \(A=A_{f, \varepsilon}\) such that \(| f(z) | \leq A e^{\varepsilon | z |}\), \(\forall z \in \mathbb{C}\), then \(f\) is called an entire function of infraexponential type. For such functions the author obtains some version of the Pólya- Levinson interpolation results. The main theorem of this paper is: Theorem 4.2. Let \(\{\lambda_ n\}^ \infty_{n=1}\) be a sequence of distinct complex numbers satisfying the following two properties: \[ \lambda_ n/n \to \infty,\;n \to \infty. \tag{1} \] (2) There is a constant \(c>0\) such that for any \(n,k \in \mathbb{N}\), \(| \lambda_ n - \lambda_ k | \geq c | n - k |\). Then there exists an entire function \(F\) of infraexponential type vanishing at all the \(\lambda_ n\), with simple zeros at a sequence \(\{\zeta_ m\}^ \infty_{m=1}\), and such that \(\forall \varepsilon > 0\), \[ 1/ \bigl | F(z) \bigr | = O(e^{\varepsilon | z |}), \quad | z - \zeta_ n | \geq (1/8)c, \quad | z | \to \infty \] \[ 1/ \bigl | F'(\zeta_ m) \bigr | = O(e^{\varepsilon | \zeta_ m |}), \quad m \to \infty. \]
entire function of infraexponential type
scientific article; zbMATH DE number 599946