Differential operators and entire functions with simple real zeros
A real entire function \(\varphi(x)\) is said to be in the \textit{Laguerre-Pólya class} if \(\varphi(x)\) can be expressed in the form \[ \varphi(x)=cx^ne^{-\alpha x^2+\beta x} \prod_{k=1}^\infty\left(1+\frac x{x_k}\right)e^{-\frac x{x_k}}, \] where \(c,\beta, x_k \in \mathbb R\), \(\alpha \geq 0\), \(n\) is a nonnegative integer and \(\sum_{k=1}^\infty 1/x_k^2 < \infty\). Let \(D:=\dfrac d{dx}\). If \(\varphi(x)=\sum_{k=0}^\infty \alpha_kx^k\) and \(f(x)\) are functions in the Laguerre-Pólya class, then under suitable hypotheses (Lemma 2) the function \(\varphi(D)f(x)=\sum_{k=0}^\infty \alpha_kf^{(k)}(x)\) also belongs to the Laguerre-Pólya class. In this well-written paper, the authors solve an open problem concerning the simplicity of the zeros of \(\varphi(D)f(x)\). In particular, they prove the following theorem. Let \(\varphi\) and \(f\) be functions in the Laguerre-Pólya class. Set \(\varphi(x)=e^{-\alpha x^2}\varphi_1(x)\) and \(f(x)= e^{-\beta x^2}f_1(x)\), where \(\varphi_1\) and \(f_1\) have genus \(0\) or \(1\), and \(\alpha, \beta\geq 0\). If \(\alpha \beta < 1/4\) and if \(\varphi\) has infinitely many zeros, then \(\varphi(D)f(x)\) has only simple real zeros.
scientific article; zbMATH DE number 2131144