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[函数] Fejér’s kernel

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hbghlyj posted 2023-5-8 06:41 |Read mode
Last edited by hbghlyj 2023-5-18 22:34
青青子衿 发表于 2013-10-26 10:23
狄利克雷核 Dirichlet Kernel
en.wikipedia.org/wiki/Dirichlet_kernel
像$\sin\theta$的求和一样
Dirichlet kernel 求和得到
\begin{equation}F_{n}(x)=\frac{1}{n+1} \sum_{k=0}^{n} D_{k}(x)=\frac{1}{n+1} \frac{\sin ^{2}[(n+1) x / 2]}{\sin ^2(x / 2)}\end{equation}
其中$D_{k}(x)=\sum_{m=-k}^{k} e^{i m x}$.

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战巡 posted 2023-5-8 14:14
\[\sum_{m=-k}^ke^{imx}=1+\sum_{m=1}^k(e^{imx}+e^{-imx})=1+2\sum_{m=1}^k\cos(mx)=\cos(kx)+\cot(\frac{x}{2})\sin(kx)\]
\[\sum_{k=0}^n\sum_{m=-k}^ke^{imx}=\sum_{k=0}^n[\cos(kx)+\cot(\frac{x}{2})\sin(kx)]\]
\[=\sum_{k=0}^n\cos(kx)+\cot(\frac{x}{2})\sum_{k=0}^m\sin(kx)\]
\[=\frac{\cos(\frac{nx}{2})\sin(\frac{n+1}{2}x)}{\sin(\frac{x}{2})}+\cot(\frac{x}{2})\cdot\frac{\sin(\frac{nx}{2})\sin(\frac{n+1}{2}x)}{\sin(\frac{x}{2})}\]
\[=\frac{[\sin(\frac{x}{2})\cos(\frac{nx}{2})+\cos(\frac{x}{2})\sin(\frac{nx}{2})]\sin(\frac{n+1}{2}x)}{\sin^2(\frac{x}{2})}\]
\[=...\]

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original poster hbghlyj posted 2023-5-19 05:33
Fejér theorem的Appendix
\begin{aligned}(n+1) F_{n}(x) & =\sum_{k=0}^{n} D_{k}(x) \\ & =\sum_{k=0}^{n} \frac{\left.\sin \left[\left(n+\frac{1}{2}\right) x\right)\right]}{\sin \left(\frac{x}{2}\right)} \\ & =\frac{1}{\sin (x / 2)} \operatorname{Im}\left\{\sum_{k=0}^{n} e^{i(k+1 / 2) x}\right\} \\ & =\frac{1}{\sin (x / 2)} \operatorname{Im}\left\{e^{i x / 2} \frac{e^{i(n+1) x}-1}{e^{i x}-1}\right\} \\ & =\frac{1}{\sin (x / 2)} \operatorname{Im}\left\{\frac{e^{i(n+1) x}-1}{e^{i x / 2}-e^{-i x / 2}}\right\} \\ & =\frac{1-\cos [(n+1) x]}{2 \sin ^{2}(x / 2)}\\&=\frac{1}{n+1} \frac{\sin ^{2}[(n+1) x / 2]}{\sin ^{2}(x / 2)}\end{aligned}

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