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Fourier Series Isometry

  1. Feb 5, 2013 #1
    Hey guys. I just started a class on Fourier Analysis and I'm having a difficult time understanding this question. Any help would be much appreciated!

    1. The problem statement, all variables and given/known data
    Verify that the Fourier Isometry holds on [−π, π] for f(t) = t. To do this, a) calculate
    the coefficients of the orthogonal Fourier series from the orthogonal series representation, b)
    calculate the sum of the squared coefficients, and c) Calculate the norm of the function as
    ∫ |f(t)|2 dt. They must be equal. How many terms in the Fourier series are necessary to have the isometry be under 5%? How many until you are under 3%, or 1%.

    2. Relevant equations
    Since f(t) = t is an odd function, you only need to calculate the sine coefficients.
    bk = 1/[itex]\sqrt{}π[/itex]∫f(t)dt


    3. The attempt at a solution
    I solved for bk and got (-2[itex]\sqrt{}π[/itex] / k)*cos(k π)
    Then I tried solving for the sum of bk ^2 but the series diverges, so now I'm stuck. Also, I'm not even sure what I'm trying to show for this problem.
     
  2. jcsd
  3. Feb 6, 2013 #2

    jbunniii

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    Science Advisor
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    Gold Member

    No, the series doesn't diverge. Note that ##\cos(k\pi) = (-1)^k##. Then (assuming your formula is correct; I haven't checked),
    $$b_k^2 = \frac{4\pi}{k^2}$$
    and so
    $$\sum b_k^2 = 4\pi \sum \frac{1}{k^2}$$
    which converges. (To what is another question.)
     
  4. Feb 6, 2013 #3
    I got it! Thank you very much for pointing that out jbunniii. I can't believe I forgot to square the k.

    Now that I've shown the Fourier Series is an isometry, I need to determine how many terms in the Fourier series are necessary to have the isometry be under 5%. I think I have to use the mean squared error formula for this but I'm not quite sure. If anyone could point me in the right direction I would really appreciate it.
     
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