The smoothest average and some extremal problems for polynomials
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| Format: | Preprint |
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2026
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| author | Gaitán, José Garzón, Carlos Madrid, José |
| author_facet | Gaitán, José Garzón, Carlos Madrid, José |
| contents | 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 $\nabla^{2k}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_25074 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The smoothest average and some extremal problems for polynomials Gaitán, José Garzón, Carlos Madrid, José Classical Analysis and ODEs 33E05, 39A12, 42A38, 65D10 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 $\nabla^{2k}$. |
| title | The smoothest average and some extremal problems for polynomials |
| topic | Classical Analysis and ODEs 33E05, 39A12, 42A38, 65D10 |
| url | https://arxiv.org/abs/2604.25074 |