The smoothest average: Dirichlet, Fejér and Chebyshev

Abstract

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 $$\underset{f\in {\ell }^2(\mathbb{Z})}{\sup }\frac{\Vert \nabla (f\ast u){\Vert }_{{\ell }^2(\mathbb{Z})}}{\Vert f{\Vert }_{{\ell }^2}}\ge \frac{2}{2n+1},$$ 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 $$\underset{f\in {\ell }^2(\mathbb{Z})}{\sup }\frac{\Vert \Delta (f\ast u){\Vert }_{{\ell }^2(\mathbb{Z})}}{\Vert f{\Vert }_{{\ell }^2(\mathbb{Z})}}\ge \frac{4}{(n+1{)}^2},$$ 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.