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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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    $\begingroup$ 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? $\endgroup$ Jan 21, 2014 at 2:22
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    $\begingroup$ Check Lebesgue -Riemann lemma. $\endgroup$ Jan 21, 2014 at 2:23
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    $\begingroup$ In addition to Mhenni's comment, note that $\cos^2(nt) = \frac 12 \left(1 + \cos(2nt)\right)$ $\endgroup$ Jan 21, 2014 at 2:28
  • $\begingroup$ @user121418 There is a variation of the lemma that holds for Fourier series (that is, holds over $[-\pi, \pi]$. Check the wiki article. $\endgroup$
    – Potato
    Jan 21, 2014 at 7:07

1 Answer 1

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

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