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Limit of a trigonometric integral

  1. Jul 5, 2012 #1
    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.
     
    Last edited: Jul 5, 2012
  2. jcsd
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