标题： Hörmander--Bernhardsson极值函数 显示英文标题

标题： The Hörmander--Bernhardsson extremal function

摘要： We characterize the function $\varphi$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $\pi$ for which $f(0)=1$. This function, studied by Hörmander and Bernhardsson in 1993, has only real zeros $\pm \tau_n$, $n=1,2, \ldots$. 基于我们之前论文中已确立的 $n+\frac12-\tau_n$ 是一个 $\ell^2$ 序列这一事实，我们以下述方式确定 $\varphi$。 我们将 $\varphi(z)$分解为 $\Phi(z)\Phi(-z)$，其中 $\Phi(z)= \prod_{n=1}^\infty(1+(-1)^n\frac{z}{\tau_n})$，并证明 $\Phi$满足某个二阶线性微分方程以及一个函数方程中的一个，这两个条件之一刻画了 $\Phi$。 我们利用这些事实，按照 Hörmander 和 Bernhardsson 的数值工作所提示的那样，建立了一个关于$n+\frac12-\tau_n$在$(n+\frac12)^{-1}$上的奇次幂级数展开式，以及$\varphi$的傅里叶变换的幂级数展开式。 $\Phi$的对偶刻画来源于一个交换关系，该关系更普遍地适用于一个双参数族的微分算子，这一事实被用于进行高精度的数值计算。 显示英文摘要

摘要： We characterize the function $\varphi$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $\pi$ for which $f(0)=1$. This function, studied by H\"{o}rmander and Bernhardsson in 1993, has only real zeros $\pm \tau_n$, $n=1,2, \ldots$. Starting from the fact that $n+\frac12-\tau_n$ is an $\ell^2$ sequence, established in an earlier paper of ours, we identify $\varphi$ in the following way. We factor $\varphi(z)$ as $\Phi(z)\Phi(-z)$, where $\Phi(z)= \prod_{n=1}^\infty(1+(-1)^n\frac{z}{\tau_n})$ and show that $\Phi$ satisfies a certain second order linear differential equation along with a functional equation either of which characterizes $\Phi$. We use these facts to establish an odd power series expansion of $n+\frac12-\tau_n$ in terms of $(n+\frac12)^{-1}$ and a power series expansion of the Fourier transform of $\varphi$, as suggested by the numerical work of H\"{o}rmander and Bernhardsson. The dual characterization of $\Phi$ arises from a commutation relation that holds more generally for a two-parameter family of differential operators, a fact that is used to perform high precision numerical computations.