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

We are interested in the ``smoothest’’ averaging that can be achieved by convolving functions $f\in {\mathrm{\ell }}^2(\mathbb{Z})$ 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 $$\underset{f\in {\mathrm{\ell }}^2(\mathbb{Z})}{\sup }\frac{\mathrm{\parallel }\mathrm{\nabla }(f\ast u){\mathrm{\parallel }}_{{\mathrm{\ell }}^2(\mathbb{Z})}}{\mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathrm{\ell }}^2}}\ge \frac{2}{2n+1},$$ and we show that equality holds if and only if $u$ is constant on the interval ${-n, \ldots, n}$\mathrm{.}Inthesettingwheresmoothnessismeasuredbythe$\ell^2$-norm of the discrete second derivative and we further restrict our attention to functions $u$ with nonnegative Fourier transform, we establish the inequality $$\underset{f\in {\mathrm{\ell }}^2(\mathbb{Z})}{\sup }\frac{\mathrm{\parallel }\mathrm{\Delta }(f\ast u){\mathrm{\parallel }}_{{\mathrm{\ell }}^2(\mathbb{Z})}}{\mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathrm{\ell }}^2(\mathbb{Z})}}\ge \frac{4}{(n+1{)}^2},$$ with equality if and only if $u$ is the triangle function $u(k)=(n+1-\mathrm{\mid }k\mathrm{\mid })\mathrm{/}(n+1{)}^2$. We also discuss a continuous analogue and several open problems.