TITLE: Find $\lim_{n\to\infty}\int_{-\pi}^{\pi}f(t)\cos^2(nt) \,dt$
QUESTION [8 upvotes]: Let $f \in C[-\pi,\pi]$.
Find the following limit:

$$\lim_{n\to\infty}\int_{-\pi}^{\pi}f(t)\cos^2(nt)\,dt$$

REPLY [2 votes]: $$
\cos^2 nx=\frac{1+\cos  2nx}{2},
$$
and hence
$$
\int_{-\pi}^\pi f(x)\,\cos^2 nx\,dx=\frac{1}{2}\int_{-\pi}^\pi f(x)\,dx+
\frac{1}{2}\int_{-\pi}^\pi f(x)\,\cos 2nx\,dx
$$
The second integral tends to zero due to Riemann-Lebesgue Lemma, and hence
$$
\lim_{n\to\infty}\int_{-\pi}^\pi f(x)\,\cos^2 nx\,dx=\frac{1}{2}\int_{-\pi}^\pi f(x)\,dx.
$$