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.
Categories: math.CA