(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Excersice of Big Rudin: ch. 4, prob. 9.

If [tex]A\subset [0,2\pi][/tex] and [tex]A[/tex] is measurable, prove that

[tex]\lim_{n\to\infty}\int_{A}\cos\,nx\,dx=0[/tex]

2. Relevant equations

Bessel's inequality

3. The attempt at a solution

I give my solution, but I post because I think that my answer is completely wrong:

[tex]\chi_{A}\in L^{2}(T)[/tex] so [tex]c_{n}=\frac{1}{2\pi}\int_{A}e^{-int}\,dt[/tex] are the Fourier coefficient and by Bessel's inequality [tex]\sum_{n=-\infty}^{\infty}|c_{n}|^{2}<\infty[/tex] therefore [tex]|c_{n}|\to 0[/tex] and [tex]\lim_{n\to\infty}\int_{A}\cos\,nx\,dx=0[/tex].

Thank you.

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# Homework Help: Limit of a trigonometric integral

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