AI 中文总结
该研究针对奇数维的尼科利斯基常数问题，构造了满足特定方程的整函数，将极值函数问题简化为有限参数谱问题，还给出了三维下高精度计算常数的方法，推广了一维相关结果。
AI 中文摘要
设φ_d(|·|)为尖锐尼科利斯基常数问题中的径向极值函数，其中常数满足C_d⁻¹=inf‖f‖₁，该下确界取自所有满足f∈PW₁¹(ℝᵈ)且f(0)=1的函数f。对于每一个奇数维，我们构造了一个指数型为1/2的整函数Φ，使得φ_d(z)=Φ(z)Φ(-z)，且Φ满足二次函数方程与带多项式系数的二阶线性微分方程。由此，求极值函数的问题被简化为含有限个参数的谱问题。该结果推广了近期的一维结果，但采用了不同的方法。例如，在三维d=3时，这一结果导出了一个七对角谱格式，可让我们高精度计算C₃。在该方法下，偶数维的情况仍未解决。
英文摘要
Let $φ_{d}(|{\,\cdot\,}|)$ be the radial extremal function in the problem for the sharp Nikolskii constant $\mathcal C_d^{-1}=\inf \|f\|_{1}$ over functions $f\in\mathit{PW}\,_{1}^{1}(\mathbb R^{d})$, $f(0)=1$. For every odd dimension, we construct an entire function $Φ$ of exponential type $1/2$ such that $φ_d(z)=Φ(z)Φ(-z)$, and $Φ$ satisfies a quadratic functional equation and a second-order linear differential equation with polynomial coefficients. Thus, the problem of finding the extremal function is reduced to a spectral problem with finitely many parameters. This result extends a recent one-dimensional result, but uses a different method. For example, in dimension $d=3$ it leads to a seven-diagonal spectral scheme that allows us to compute $\mathcal C_3$ to high accuracy. Even dimensions remain open within this approach.
Comments31 pages
