Let $f \in C[-\pi,\pi]$.
Find the following limit:
$$\lim_{n\to\infty}\int_{-\pi}^{\pi}f(t)\cos^2(nt)\,dt$$
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1$\begingroup$ 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? $\endgroup$– Ben GrossmannJan 21, 2014 at 2:22
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1$\begingroup$ Check Lebesgue -Riemann lemma. $\endgroup$– Mhenni BenghorbalJan 21, 2014 at 2:23
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5$\begingroup$ In addition to Mhenni's comment, note that $\cos^2(nt) = \frac 12 \left(1 + \cos(2nt)\right)$ $\endgroup$– Ben GrossmannJan 21, 2014 at 2:28
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$\begingroup$ @user121418 There is a variation of the lemma that holds for Fourier series (that is, holds over $[-\pi, \pi]$. Check the wiki article. $\endgroup$– PotatoJan 21, 2014 at 7:07
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1 Answer
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$$ \cos^2 nx=\frac{1+\cos 2nx}{2}, $$ 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 $$ \lim_{n\to\infty}\int_{-\pi}^\pi f(x)\,\cos^2 nx\,dx=\frac{1}{2}\int_{-\pi}^\pi f(x)\,dx. $$
answered Jan 21, 2014 at 21:15