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

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

Find the following limit:

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

• What are your thoughts? Have you tried anything yet? Is this from class/for homework? If so, are there any tricks you went over recently that might apply here? Commented Jan 21, 2014 at 2:22
• Check Lebesgue -Riemann lemma. Commented Jan 21, 2014 at 2:23
• In addition to Mhenni's comment, note that $\cos^2(nt) = \frac 12 \left(1 + \cos(2nt)\right)$ Commented Jan 21, 2014 at 2:28
• @user121418 There is a variation of the lemma that holds for Fourier series (that is, holds over $[-\pi, \pi]$. Check the wiki article. Commented Jan 21, 2014 at 7:07

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