The smoothest average and some extremal problems for polynomials

Abstract

We study the problem of finding the "smoothest'' local average of a function $f \in \ell^2(\mathbb{Z})$ when we consider its convolution with suitable kernels $u$. The measurement of smoothness is as follows: Given a positive integer $k$, we aim to minimize the constant \begin{equation*} \sup_{0 \neq f \in \ell^2(\mathbb{Z})} \frac{\|\nabla^{k}(u\ast f)\|_{\ell^2(\mathbb{Z})}}{\|f\|_{\ell^2(\mathbb{Z})}} \end{equation*} among all symmetric kernels $u : \{-n,\dots,n\} \to \mathbb{R}$ with normalization $\sum_{j=-n}^{n}u(j) = 1$. We are also interested in finding the kernel for which the least constant is attained. For $k=1$ and $k=2$, the sharp constants and optimal kernels were obtained by Kravitz-Steinerberger, and Richardson. In this paper, we provide alternative proofs for $k\in \{1,2\}$ by using complex analysis tools. Moreover, we establish the case $k=3$, and also the cases $k\in \{4,6\}$ when the kernels are restricted to have non-negative Fourier transform. These are the first results in the literature for $k>2$. Finally, we deduce a general relation between the sharp constants and optimal kernels corresponding to $\nabla^k$ and $\nabla^{2k}$.