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Math Help - A question on Fourier series

  1. #1
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    A question on Fourier series

    I need to know an example of a function defined on the interval [0, pi] which cannot be expanded as a Fourier series. Thanks.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by curvature View Post
    I need to know an example of a function defined on the interval [0, pi] which cannot be expanded as a Fourier series. Thanks.
    How about 1/x^2.

    In particular how would you evaluate:

     \int_0^{\pi} \frac{\cos(2nx)}{x^2}~dx

    RonL
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    Thanks. But I was thinking of an example of function with infinitely many discontinuities or with infinitely many extrema, if any.
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  4. #4
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    Quote Originally Posted by curvature View Post
    Thanks. But I was thinking of an example of function with infinitely many discontinuities or with infinitely many extrema, if any.
    f(x) = \left\{ \begin{array}{c}0\mbox{ if }x\in \mathbb{Q} \\ 1\mbox{ if }x\not \in \mathbb{Q}\end{array} \right..
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker View Post
    f(x) = \left\{ \begin{array}{c}0\mbox{ if }x\in \mathbb{Q} \\ 1\mbox{ if }x\not \in \mathbb{Q}\end{array} \right..
    A Fourier series converges to the function a.e. so the zero Fourier series converges to this a.e. as \mathbb{Q} is a null set. That is this function does have a Fourier series and it converges to the function a.e. as required.

    (even the Fourier series of a function with a step discontinuity does not converge to the function at the step without a bit of jiggery pokery about what the value of the function is at the step)

    RonL
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  6. #6
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    Quote Originally Posted by ThePerfectHacker View Post
    f(x) = \left\{ \begin{array}{c}0\mbox{ if }x\in \mathbb{Q} \\ 1\mbox{ if }x\not \in \mathbb{Q}\end{array} \right..
    Thanks. I know the function is called the Dirichlet function and I guess this bounded function cannot be expressed into a Fourier series, but not quite sure since no books I read tell the fact.

    I wonder if examples like this motivated Dirichlet to propose the Dirichlet Conditon for Fourier series?
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