Computation of an integral about Fejér kernel

Question

How can I compute the integral?

$$\lim_ {n \rightarrow \infty}\int_{0}^{\pi/4}{ \frac { \sin^2(\frac{nx}{2})}{n\,\sin^2(\frac{x}{2})} }\,dx$$

I think it is Fejér kernel application, but I have no idea how to do.

Can someone give me details?

Answer 1

Observe \begin{align} \int^\pi_{-\pi} F_n(x)\ dx = 2\pi \end{align} where $F_n(x)$ is the Fejer kernel.

Moreover, we have that \begin{align} \int^\pi_{-\pi} F_n(x)\ dx = \int^{\pi/4}_{-\pi/4} F_n(x)\ dx + \int_{\pi>|x|>\pi/4} F_n(x)\ dx \end{align} where \begin{align} \lim_{n\rightarrow \infty} \int_{\pi>|x|>\pi/4} F_n(x)\ dx = 0. \end{align} Hence it follows \begin{align} \lim_{n\rightarrow\infty} \int^{\pi/4}_{-\pi/4} F_n(x)\ dx = 2\pi. \end{align} Since $F_n(x)$ is even, then it follows \begin{align} \lim_{n\rightarrow \infty}\int^{\pi/4}_0 F_n(x)\ dx = \pi. \end{align}

Answer 2

Back to the sum definition of the Fejér kernel: $$ F_n(x) = \frac{1}{n}\left(\frac{\sin^2\frac{nx}{2}}{\sin^2\frac{x}{2}}\right) = \frac{1}{n}\sum_{k=0}^{n-1}\left(1+2\sum_{j=1}^{k}\cos(jx)\right)\tag{1}$$ We may now apply termwise integration: $$\int_{0}^{\pi/4}F_n(x)\,dx = \frac{\pi}{4}+\frac{2}{n}\sum_{k=1}^{n-1}\sum_{j=1}^{k}\frac{\sin\frac{\pi j}{4}}{j}=\frac{\pi}{4}+\frac{2}{n}\sum_{j=1}^{n-1}\frac{n-j}{j}\sin\frac{\pi j}{4}\tag{2}$$ and since $\sum_{j=1}^{m}\sin\frac{\pi j}{4}$ is bounded, $$ \lim_{n\to +\infty}\int_{0}^{\pi/4}F_n(x)\,dx = \frac{\pi}{4}+2\sum_{j\geq 1}\frac{\sin\frac{\pi j}{4}}{j}=\frac{\pi}{4}+\frac{3\pi}{4}=\color{red}{\large \pi}.\tag{3} $$

Answer 3

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Without the Fej$\acute{e}$r Kernel !!!.

$\bbx{\ds{\mbox{As}\ n \to \infty}}$: \begin{align} \int_{0}^{\pi/4}{\sin^{2}\pars{nx/2} \over n\sin^2\pars{x/2}}\,\dd x & \sim {1 \over n}\,n^{2}\int_{0}^{\pi/4}{\sin^{2}\pars{nx/2} \over \pars{nx/2}^{2}}\,\dd x = n\,{2 \over n}\int_{0}^{n\pi/8}{\sin^{2}\pars{x} \over x^{2}}\,\dd x \\[5mm] & \sim 2\ \underbrace{\int_{0}^{\infty}{\sin^{2}\pars{x} \over x^{2}}\,\dd x} _{\ds{\pi \over 2}} = \bbox[#ffe,10px,border:1px dotted navy]{\ds{\pi}} \end{align}