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I’m having trouble showing that the $L^p$ norm of the $n$-th order Dirichlet kernel is proportional to $n^{1-1/p}$.

I’ve tried brute force integration and it didn’t work out. I would be grateful for any hints.

Thanks.

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Assuming the notation of DJC, apply the transformation $y = n x$ to obtain

$$

\frac{1}{n^{p-1}} \left\| D_n \right\|^p_p = \int_{- n \pi}^{n \pi} \left| \frac{\sin(y + \frac{y}{2 n})}{n \sin(\frac{y}{2 n})} \right|^p dy .

$$

Fatou’s Lemma now shows that

$$

\frac{1}{n^{p-1}} \left\| D_n \right\|^p_p \leq \int_{\mathbb{R}} \left| \frac{2 \sin(y)}{y} \right|^p dy .

$$

For the other side of the inequality, note, that

$$

\frac{1}{n^{p-1}} \left\| D_n \right\|^p_p \geq \int_{- n \pi}^{n \pi} \left| \frac{\sin(y + \frac{y}{2 n})}{\frac{y}{2}} \right|^p dy,

$$

so using the triangular inequality of the $L^p$ norm we obtain

$$

\frac{1}{n^{p-1}} \left\| D_n \right\|^p_p \geq \left( \left( \int_{\mathbb{R}} \left| \frac{2 \sin(y)}{y} \right|^p dy \right)^{1/p} – \left( \int_{- n \pi}^{n \pi} \left| \frac{\sin(y + \frac{y}{2 n}) – \sin y}{\frac{y}{2}} \right|^p dy \right)^{1/p} \right)^p .

$$

The inequality now follows from

$$

\int_{- n \pi}^{n \pi} \left| \frac{\sin(y + \frac{y}{2 n}) – \sin y}{\frac{y}{2}} \right|^p dy \leq \int_{- n \pi}^{n \pi} \left| \frac{\frac{y}{2 n}}{\frac{y}{2}} \right|^p dy = \frac{2 \pi}{n^{p – 1}} = o(n) .

$$

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