高维空间中的Hörmander–Bernhardsson函数
The Hörmander--Bernhardsson function in higher dimensions
AI总结：
该研究在高维Paley–Wiener空间中求点赋值算子范数，证明径向极值函数满足三阶线性ODE，推广了奇数维结果，还证明了相关插值与倒数公式。
AI中文摘要：
我们研究在Paley–Wiener空间$PW^{1}(\mathbb{R}^d)$中求点赋值算子范数的问题，该空间由$\mathbb{R}^d$上可积的d变量球面指数型函数构成。极值函数可取径向形式，这自然引导我们考虑单变量函数的加权Paley–Wiener空间中的相关极值问题。我们证明对任意$d \geq 1$，径向极值函数必满足带多项式系数的三阶线性常微分方程，推广了戈尔巴乔夫（Gorbachev）近期的奇数维结果。在此过程中，我们还证明了涉及极值函数零点的插值公式与倒数公式。
英文摘要：
We study the problem of finding the norm of the point evaluation operator in the Paley--Wiener space $PW^{1}(\r^d)$, consisting of $d$-variable functions of spherical exponential type that are integrable on $\r^d$. The extremal functions can be taken radial, which naturally leads us to consider a related extremal problem in a weighted Paley--Wiener space of single-variable functions. We establish that the radial extremal function must satisfy a third-order linear ODE with polynomial coefficients for every $d \geq 1$, extending Gorbachev's recent odd-dimensional result. Along the way, we prove interpolation and reciprocal formulas involving the zeros of the extremizer.
