The smoothest average: Dirichlet, Fej\'er and Chebyshev

We are interested in the ``smoothest'' averaging that can be achieved by convolving functions $f \in \ell^2(\mathbb{Z})$ with an averaging function $u$. More precisely, suppose $u:\{-n, \ldots, n\} \to \mathbb{R}$ is a symmetric function normalized to $\sum_{k=-n}^{n}u(k) = 1$. We show that every convolution operator is not-too-smooth, in the sense that $$\sup_{f \in \ell^2(\mathbb{Z})} \frac{\| \nabla (f*u)\|_{\ell^2(\mathbb{Z})}}{\|f\|_{\ell^2}}\geq \frac{2}{2n+1},$$ and we show that equality holds if and only if $u$ is constant on the interval $\{-n, \ldots, n\}$. In the setting where smoothness is measured by the $\ell^2$-norm of the discrete second derivative and we further restrict our attention to functions $u$ with nonnegative Fourier transform, we establish the inequality $$\sup_{f \in \ell^2(\mathbb{Z})} \frac{\| \Delta (f*u)\|_{\ell^2(\mathbb{Z})}}{\|f\|_{\ell^2(\mathbb{Z})}} \geq \frac{4}{(n+1)^2},$$ with equality if and only if $u$ is the triangle function $u(k)=(n+1-|k|)/(n+1)^2$. We also discuss a continuous analogue and several open problems.