A generalization of the Laguerre-Pólya class of entire functions (Q1960898)

A classical result of Laguerre and Pólya asserts that a real entire function \(f\) can be uniformly approximated on bounded subsets of \(\mathbb C\) by polynomials having only real zeros if and only if f(z) can be expressed in the form \[ f(z)=c z^m e^{\alpha z-\gamma z^2}\prod_{k=1}^\omega \left(1+ \frac z{x_k}\right)e^{-\frac z{x_k}}, \qquad (1\leq \omega \leq \infty), \] where \(c,\alpha, x_k \in \mathbb R\), \(c, x_k \neq 0\), \(\gamma \geq 0\), \(m\) is a nonnegative integer and \(\sum_{k=1}^\infty 1/x_k^2 < \infty\). The purpose of the paper under review is to generalize this theorem. Let \(B\) be a finite set of positive integers and let \(\Theta\) denote a set of real numbers which is unbounded on both sides. The author characterizes the entire functions that can be uniformly approximated on bounded sets by polynomials of the form \(p(z)=\prod_{j \in B}p_j(z^j)\), where \(p_j(z)\) is a polynomial all of whose zeros lie in \(\Theta\).