论文标题
平均平均水平：Dirichlet，Fejér和Chebyshev
The smoothest average: Dirichlet, Fejér and Chebyshev
论文作者
论文摘要
我们对可以通过平均功能$ u $在\ ell^2（\ mathbb {z}）$ in \ ell^2（\ mathbb {z}）$中的函数$ f \来实现的``最平稳''平均感兴趣。更准确地说，假设$ u：\ { - n，\ ldots，n \} \ to \ mathbb {r} $是归一化为$ \ sum_ {k = -n}^n}^{n} u（k）= 1 $的对称函数。我们表明，每个卷积操作员都不平滑，从某种意义上说，$ \ sup_ {f \ in \ ell^2（\ mathbb {z}）} \ frac {\ | \ nabla（f*u）\ | _ {\ ell^2（\ Mathbb {z}）}}}}}} {\ | f \ | _ {\ | _ {\ ell^2}} \ geq \ geq \ frac {2} {2} {2n+1} \ ldots，n \} $。在通过$ \ ell^2 $ -Norm测量离散第二个衍生物的情况下，我们将注意力限制为具有非负傅立叶变换的函数$ u $，我们建立了不平等$$ \ sup_ {f \ in \ ell^in \ ell^2（\ mathbb {z}}}}} \ frac \ frac \ frac \ frac \ frac { δ（f*u）\ | _ {\ ell^2（\ Mathbb {z}）}}}}} {\ | f \ | _ {\ ell^2（\ ell^2（\ MathBB {z}}}}}} \ geq \ geq \ geq \ geq \ frac {4} {4} {4} {（n+1）{（n+1） $ u（k）=（n+1- | k |）/（n+1）^2 $。我们还讨论了一个连续的类似物和几个开放问题。
We are interested in the ``smoothest'' averaging that can be achieved by convolving functions $f \in \ell^2(\mathbb{Z})$ with an averaging function $u$. More precisely, suppose $u:\{-n, \ldots, n\} \to \mathbb{R}$ is a symmetric function normalized to $\sum_{k=-n}^{n}u(k) = 1$. We show that every convolution operator is not-too-smooth, in the sense that $$\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.
