We consider functions $f: \mathbb{Z} \to \mathbb{R}$ and kernels $u: \{-n, \cdots, n\} \to \mathbb{R}$ normalized by $\sum_{\ell = -n}^{n} u(\ell) = 1$, making the convolution $u \ast f$ a "smoother" local average of $f$. We identify which choice of $u$ most effectively smooths the second derivative in the following sense. For each $u$, basic Fourier analysis implies there is a constant $C(u)$ so $\|\Delta(u \ast f)\|_{\ell^2(\mathbb{Z})} \leq C(u)\|f\|_{\ell^2(\mathbb{Z})}$ for all $f: \mathbb{Z} \to \mathbb{R}$. By compactness, there is some $u$ that minimizes $C(u)$ and in this paper, we find explicit expressions for both this minimal $C(u)$ and the minimizing kernel $u$ for every $n$. The minimizing kernel is remarkably close to the Epanechnikov kernel in Statistics. This solves a problem of Kravitz-Steinerberger and an extremal problem for polynomials is solved as a byproduct.

翻译：考虑函数 $f: \mathbb{Z} \to \mathbb{R}$ 与核函数 $u: \{-n, \cdots, n\} \to \mathbb{R}$，且满足归一化条件 $\sum_{\ell = -n}^{n} u(\ell) = 1$，使得卷积 $u \ast f$ 成为 $f$ 的"更平滑"局部平均值。我们确定在如下意义上，哪种 $u$ 的选择能最有效地平滑二阶导数。对每个 $u$，基础傅里叶分析表明存在常数 $C(u)$，使得对所有 $f: \mathbb{Z} \to \mathbb{R}$ 有 $\|\Delta(u \ast f)\|_{\ell^2(\mathbb{Z})} \leq C(u)\|f\|_{\ell^2(\mathbb{Z})}$。由紧性知，存在某个 $u$ 使 $C(u)$ 最小化，本文针对每个 $n$ 给出了该最小 $C(u)$ 及最小化核函数 $u$ 的显式表达式。该最小化核函数与统计学中的埃帕涅奇尼科夫核高度接近。这解决了Kravitz-Steinerberger提出的一个问题，并作为副产品解决了一个多项式极值问题。