Based on the explicit formula of the pointwise error of Fourier projection approximation and by applying van der Corput-type Lemma, optimal convergence rates for periodic functions with different degrees of smoothness are established. It shows that the convergence rate enjoys a decay rate one order higher in the smooth parts than that at the singularities. In addition, it also depends on the distance from the singularities. Ample numerical experiments illustrate the perfect coincidence with the estimates.

中文翻译：

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有限正则函数的傅里叶展开的局部行为

基于傅里叶投影近似逐点误差的显式公式，应用van der Corput型引理，建立了不同平滑度周期函数的最优收敛速率。结果表明，平滑部分的收敛速度比奇点处的衰减速度高一阶。此外，它还取决于距奇点的距离。大量的数值实验表明与估计值完全一致。