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Fourier series: Parseval's identity HELP!

  1. Oct 13, 2006 #1
    Hey all,

    I am unsure how to do this problem... i find problems where i have to derive things quite difficult! :P

    [​IMG]

    this is the Full Fourier series i think and so the Fourier coeffiecients would be given by:

    [​IMG]

    ok so first i need to take the inner product, so i did this:

    [​IMG]

    but then i am stuck... anyone got an idea of how to proceed from here?

    Cheers! :)

    Sarah
     
  2. jcsd
  3. Oct 13, 2006 #2

    quasar987

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    How about writing the inner product in this form instead...

    [tex]\int_{-L}^{+L}|f(x)|^2dx = \int_{-L}^{+L} f(x) \left( \frac{A_0}{2} + \sum_{n\geq 1} A_n\cos (\frac{n\pi x}{L})+B_n\sin (\frac{n\pi x}{L}) \right)dx[/tex]

    does this inspire you more? (It should)
     
  4. Oct 13, 2006 #3
    it does actually, hold on.... i'll post my answer in a sec
     
  5. Oct 13, 2006 #4
    [​IMG]

    however i do have one question, are we allowed to substitute the integral inside the summation? (i am have never heard of any rules telling me whether this is or is not allowed....)
     
  6. Oct 13, 2006 #5

    quasar987

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    Haven't you covered series of functions (or at least sequences of functions) in an earlier class?
     
    Last edited: Oct 13, 2006
  7. Oct 13, 2006 #6
    not for a couple of years i don't think...

    thanks for the help by the way :)
     
  8. Oct 13, 2006 #7

    quasar987

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    Strange, since Fourier series are precisely series of functions. In occurence, sine and cosine. They are usually covered in a second analysis class together with the theory of Riemann integration.
     
  9. Oct 13, 2006 #8
    yeah, we did Riemann integration about a year and half ago i think, in a first analysis class i'm pretty sure.
     
  10. Dec 2, 2006 #9
    So when is it OK to move the integral inside the summation?
     
  11. Dec 2, 2006 #10

    quasar987

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    Thm: If the series of function [itex]\sum_{n=1}^{\infty} f_n(x)[/itex] converges uniformly towards S(x) on [a,b] and if [itex]f_n[/itex] is integrable on [a,b] [itex]\forall n\in \mathbb{N}[/itex], then the function S is integrable on [a,b] and

    [tex]\sum_{n=1}^{\infty}\int_a^b f_n(x)dx=\int_a^b \sum_{n=1}^{\infty} f_n(x)dx[/tex]
     
  12. Mar 17, 2008 #11
    In this case it is surely ok to include the integral in the summation. Don't know the rule exactly but I remember my professor claiming that it's ok to do the above one.

    I will try to find supporting evidence in due course!
     
  13. May 6, 2011 #12
    In this case, how do we check if the Fourier series converges uniformly to the f(x)?
    Actually we don't know if the function f is continuous and C^1 in [-L,L] so I think that we can't "move the integral inside the summation". If it was saying, for example, that the function f is periodic with L=pi and f is square integrable than we could do this.
     
    Last edited: May 6, 2011
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