Hello,

I am trying to prove some problem out of Rudin, chapter 8, problem 12, part d). In particular,

Suppose $0<\delta< \pi$, $f(x)=1$ if $|x|<\delta$, $f(x)=0$ if $\delta<|x|\leq \pi$, and $f(x+2\pi)=f(x)$ for all $x$.

You need part c) before the question itself makes sense,

c) Deduce from Parseval's theorem that $$\sum_{n=1}^\infty \frac{\sin^2(n\delta)}{n^2\delta}=\frac{\pi-\delta}{2}.$$

The proof of c) is not so bad.

Parseval's theorem, specifically line (85) on page 191 says that: $$\frac{1}{2\pi}\int_{-\delta}^{\delta}|f(x)|^2 dx=\sum_{-\infty}^{\infty}|c_m|^2$$ We can rewrite this using the real form of the Fourier series, i.e., if $f(x)\sim \frac{a_0}{2}+\sum_{m=1}^\infty(a_m\cos(mx)+b_m\si n(mx))$, then $$\frac{1}{\pi}\int_{-\delta}^{\delta}f^2(x)dx=\frac{a_0^2}{2}+\sum_{m=1 }^\infty(a_m^2+b_m^2)$$ Here, since the $b_m$s are zero we have, $$\frac{1}{\pi}\int_{-\delta}^{\delta}f^2(x)dx=\frac{a_0^2}{2}+\sum_{m=1 }^\infty(a_m^2)$$ Hence, $$\frac{2\delta}{\pi}=\frac{1}{\pi}\int_{-\delta}^{\delta}f^2(x)dx=\frac{a_0^2}{2}+\sum_{m=1 }^\infty (a_m^2)=\frac{4\delta^2}{2\pi^2}+\sum_{m=1}^\infty \Big(\frac{2\sin(m\delta)}{\pi m}\Big)^2=\frac{\delta^2}{2\pi^2}+\sum_{m=1}^{\inf ty} \frac{2^2\sin^2(m\delta)}{\pi^2 m^2}$$ Which we can rewrite as, \begin{equation*}\begin{split}\frac{2\delta}{\pi}& =\frac{4\delta^2}{2\pi^2}+\sum_{m=1}^\infty \frac{4\sin^2(m\delta)}{\pi^2m^2}\\ \frac{\pi^2}{4\delta}\frac{2\delta}{\pi}&=\frac{4\ delta^2}{2\pi^2}\frac{\pi^2}{4\delta}+\sum_{m=1}^\ infty \frac{4\sin^2(m\delta)}{\pi^2m^2}\frac{\pi^2}{4\de lta}\\ \frac{\pi}{2}&=\frac{\delta}{2}+\sum_{m=1}^\infty \frac{\sin^2(m\delta)}{\delta m^2}\\ \frac{\pi-\delta}{2}&=\sum_{m=1}^\infty \frac{\sin^2(m\delta)}{\delta m^2}\end{split}\end{equation*}

d) Let $\delta \to 0$ and prove that $$\int_0^\infty \Big(\frac{\sin x}{x}\Big)^2 dx = \frac{\pi}{2}.$$

The idea is supposed to be to let the partial sums of this integral be the LHS of the equality proved in part c), then hopefully blah blah blah my way to replacing something with $\frac{\pi}{2}$. I'm having trouble justifying this or working at all with this stuff. I should have to evaluate

$$\lim_{t \to \infty} \int_0^t \Big(\frac{\sin x}{x}\Big)^2 dx$$

which has Riemann sums, $\sum_{i=1}^n \Big(\frac{\sin(nt/n)}{\sqrt{nt/n}}\Big)^2\Delta(t)$?

So I'm not really sure how to justify this, and even then what to do next.