Let $$D_N(x)=\frac{\sin [(N+(1/2))t]}{\sin (t/2)}$$ be the Dirichlet kernel. Let $x(N)$ be the number in $0
A limit related to the Gibbs phenomenon
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analysis
fourier-analysis
fourier-series
1 Answers
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Looks true to me.
$\displaystyle x(N) = \dfrac{\pi}{N+1}$
Now $\displaystyle D_n(x) = 1 + 2\sum_{k=1}^{n} \cos(kx)$
and $\displaystyle |\sin x - \sin y| \leq |x-y|$ (easily seen using Mean Value Theorem)
Thus
$$\displaystyle \left|\int_{\pi/(N+1)}^{\pi/N} D_N(t) \ \text{dt}\right| = \left|\int_{\pi/(N+1)}^{\pi/N} 1 + 2\sum_{k=1}^{N} \cos(kx) \ \text{dt}\right|$$
$$\displaystyle = \frac{\pi}{N(N+1)} + 2 \sum_{k=1}^{N} \mathcal{O}(\frac{1}{N^2}) = \mathcal{O}(\frac{1}{N})$$
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0Very nice. I did not see that $x(N)$ can be so simple!! – 2010-12-05