AI 中文总结
该研究针对任意维度d≥1的PW₁¹(ℝᵈ)，通过分解径向极值函数φ，将原极值问题简化为一维谱问题，还得到相关多项式系数的zeta解释与零点平衡条件，可用于构建计算Nikolskii常数𝒞_d的算法。
AI 中文摘要
我们研究径向极值函数φ(|·|)，该函数源于对任意维度d≥1的PW₁¹(ℝᵈ)中精确Nikolskii常数𝒞_d的求解问题。我们证明了φ可分解为φ=Φ₁Φ₂，其中Φ₁和Φ₂是指数型为1/2的整函数，满足函数方程及带多项式系数的二阶微分方程。由此，原极值问题被简化为一个最多依赖d+1个参数的一维谱问题。我们还得到了函数方程中多项式系数的zeta解释，以及极值函数零点的乘性平衡条件。这些结果可用于构建多个计算𝒞_d的算法。
英文摘要
We study the radial extremal function $φ(|{\,\cdot\,}|)$ arising in the problem of finding the sharp Nikolskii constant $\mathcal C_d$ in $\mathit{PW}_1^1(\mathbb R^d)$ for arbitrary dimension $d\ge1$. We prove a factorization $φ=Φ_1Φ_2$, where $Φ_1$ and $Φ_2$ are entire functions of exponential type $1/2$ satisfying a functional equation and second-order differential equations with polynomial coefficients. As a result, the original extremal problem is reduced to a one-dimensional spectral problem depending on at most $d+1$ parameters. We also obtain a zeta interpretation of the coefficients of the polynomial appearing in the functional equation and a multiplicative equilibrium condition for the zeros of the extremal function. These results can be used to construct several algorithms for computing $\mathcal C_d$.
Comments22 pages
