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

Question

Let $f \in C[-\pi,\pi]$.

Find the following limit:

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

Answer 1

$$ \cos^2 nx=\frac{1+\cos 2nx}{2}, $$ 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 $$ \lim_{n\to\infty}\int_{-\pi}^\pi f(x)\,\cos^2 nx\,dx=\frac{1}{2}\int_{-\pi}^\pi f(x)\,dx. $$