  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.
