Summary: 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 \(u : \{-n, \dots, n\} \to \mathbb{R}\) 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 \[ \sup_{f \in \ell^2 (\mathbb{Z})} \frac{\Vert\nabla (f \ast u) \Vert_{\ell^2 (\mathbb{Z})}}{\Vert f \Vert_{\ell^2}} \geqslant \frac{2}{2n + 1}, \] and we show that equality holds if and only if \(u\) is constant on the interval \(\{-n, \dots, n\}\). 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 \[ \sup_{f \in \ell^2 (\mathbb{Z})} \frac{\Vert \Delta (f \ast u) \Vert_{\ell^2 (\mathbb{Z})}}{\Vert f \Vert_{\ell^2 (\mathbb{Z})}} \geqslant \frac{4}{(n + 1)^2}, \] 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.