基于欧拉和加速的傅里叶级数恢复指数（几何）收敛速度的证明，针对具有吉布斯现象的函数
A Proof, Based on the Euler Sum Acceleration, of the Recovery of an Exponential (Geometric) Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon
AI总结：
针对具有奇点的函数，通过欧拉加速（等价于共形映射）证明傅里叶级数在远离奇点处可恢复指数收敛速度，并纠正了关于仅能恢复根指数收敛的错误说法。
摘要：
当函数 $f(x)$ 在实轴上的点 $x_{s}$ 处具有奇点时，其傅里叶级数截断至第 $N$ 项后，在整个实轴上的逐点误差仅为 $O(1/N)$。这种奇点在气象学和海洋学中作为“锋面”，在流体力学其他分支中作为“激波”自发出现。先前研究表明，在远离奇点的所有点上，可以恢复指数收敛速度，即 $|f(x) - f_{N}^σ(x)| \sim O(\exp(- q(x) N))$，其中 $f_{N}^σ(x)$ 是对部分和 $f_{N}(x)$ 应用滤波或可和法的结果，$q(x)$ 是与 $x$ 到奇点距离 $d(x) \equiv |x-x_{s}|$ 相关的比例常数。本文通过虚拟变量 $z$ 的共形映射给出了一个具有广泛性的初等证明；这等价于应用欧拉加速。我们证明，对于傅里叶周期为 $2π$ 的欧拉滤波器，$q(x) \approx \log(\cos(d(x)/2))$。更复杂的滤波器可以增大 $q(x)$，但欧拉滤波器最为简单。我们还能纠正近期发表的声称对于紧支撑滤波器（如欧拉加速和 Erfc-Log 滤波器）仅能恢复根指数收敛速度的说法。
英文摘要：
When a function $f(x)$ is singular at a point $x_{s}$ on the real axis, its Fourier series, when truncated at the $N$-th term, gives a pointwise error of only $O(1/N)$ over the entire real axis. Such singularities spontaneously arise as "fronts" in meteorology and oceanography and "shocks" in other branches of fluid mechanics. It has been previously shown that it is possible to recover an exponential rate of convegence at all points away from the singularity in the sense that $|f(x) - f_{N}^σ(x) | \sim O(\exp(- q(x) N))$ where $f_{N}^σ(x)$ is the result of applying a filter or summability method to the partial sum $f_{N}(x)$ and $q(x)$ is a proportionality constant that is a function of $d(x) \equiv |x-x_{s}|$, the distance from $x$ to the singularity. Here we give an elementary proof of great generality using conformal mapping in a dummy variable $z$; this is equivalent to applying the Euler acceleration. We show that $q(x) \approx \log(\cos(d(x)/2))$ for the Euler filter when the Fourier period is $2 π$. More sophisticated filters can increase $q(x)$, but the Euler filter is simplest. We can also correct recently published claims that only a root-exponential rate of convergence can be recovered for filters of compact support such as the Euler acceleration and the Erfc-Log filter.
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。
