最平滑的平均值：狄利克雷、费耶尔和切比雪夫
The smoothest average: Dirichlet, Fejér and Chebyshev
摘要：
我们感兴趣的是“最平滑”的平均，可以通过卷积函数$f \in \ell^2(\mathbb{Z})$与平均函数$u$来实现。更精确地，假设$u:\{-n, \ldots, n\} \to \mathbb{R}$是归一化为$\sum_{k=-n}^{n}u(k) = 1$的对称函数。我们表明，每个卷积算子是不太光滑，在这个意义上，$$\sup_{f \in \ell^2(\mathbb{Z})} \frac{\| \nabla (f*u)\|_{\ell^2(\mathbb{Z})}}{\|f\|_{\ell^2}}\geq \frac{2}{2n+1},$$，我们表明，平等保持当且仅当$u$是恒定的区间$\{-n, \ldots, n\}$。在设置中的光滑性是衡量的$\ell^2$ -范数的离散二阶导数，我们进一步限制我们的注意力与非负傅立叶变换的功能$u$，我们建立不等式$$\sup_{f \in \ell^2(\mathbb{Z})} \frac{\| Δ(f*u)\|_{\ell^2(\mathbb{Z})}}{\|f\|_{\ell^2(\mathbb{Z})}} \geq \frac{4}{(n+1)^2},$$与平等的当且仅当$u$是三角函数$u(k)=(n+1-|k|)/(n+1)^2$。我们还讨论了连续模拟和几个开放的问题。
英文摘要：
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 $$\sup_{f \in \ell^2(\mathbb{Z})} \frac{\| Δ(f*u)\|_{\ell^2(\mathbb{Z})}}{\|f\|_{\ell^2(\mathbb{Z})}} \geq \frac{4}{(n+1)^2},$$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
