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Math Help - Question on Riemann integration

  1. #1
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    Question on Riemann integration

    Suppose that f is Riemann integrable on (-\pi,\pi ). Prove that:
    \lim_{n \to \infty} \int_{-\pi}^{\pi} \! f(x)cos(nx)dx=0.
    I can't see how to start this question, could someone help?
    Thanks.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by worc3247 View Post
    Suppose that f is Riemann integrable on (-\pi,\pi ). Prove that:
    \lim_{n \to \infty} \int_{-\pi}^{\pi} \! f(x)cos(nx)dx=0.
    I can't see how to start this question, could someone help?
    Thanks.
    Hint:

    Integration by parts.

    \cos(nx)=(\frac{sin(nx)}{n})'
    Last edited by Also sprach Zarathustra; May 30th 2011 at 04:16 AM.
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  3. #3
    Super Member girdav's Avatar
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    Show the result when f is a simple function (a function which only takes a finite number of values).
    Last edited by girdav; May 30th 2011 at 04:13 AM.
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  4. #4
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    Quote Originally Posted by worc3247 View Post
    Suppose that f is Riemann integrable on (-\pi,\pi ). Prove that:
    \lim_{n \to \infty} \int_{-\pi}^{\pi} \! f(x)cos(nx)dx=0.
    I can't see how to start this question, could someone help?
    Thanks.
    This result is known as the Riemann–Lebesgue lemma. If the function f is differentiable, then Also sprach Zarathustra's hint provides a quick and easy proof. If f is continuous then the R–L lemma is an easy consequence of Fejér's theorem. But if all you know about f is that it is Riemann integrable, then you have to follow girdav's advice and approximate f by a simple function.

    The R–L lemma is easy to prove for the characteristic function of an interval (by explicitly calculating the integral). The result then follows for a simple function (which is just a finite linear combination of characteristic functions). Finally, to approximate a Riemann integrable function by a simple function, take a Riemann sum corresponding to a suitably fine dissection of the interval.
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