Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The smoothest average: Dirichlet, Fej\'er and Chebyshev   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.

17
The categories of this paper are: q-fin
16
The categories of this paper are: stat
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio