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Math Help - fourrier transform proof

  1. #1
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    fourrier transform proof

    is there a proof 4 fourrier's theory;that any signal is composed of infinite number of sinusoids?

    another question : what is the reason that made fourrier think of signals in that way? i mean , of course he faced some problem or had a certain idea before he put his theory
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by allah's_slave View Post
    is there a proof 4 fourrier's theory;that any signal is composed of infinite number of sinusoids?

    For a continuous absolutely integrable function f(x) with absolutly integrable Fourier transform [\mathcal{F}f](\xi)=F(\xi) the result follows from:

    f(0)=\int_{-\infty}^{\infty} F(\xi)~d\xi \ \ \ \ \ \ \  ...(1)

    by the translation theorem.

    To prove (1) you need a sequence of well behaved functions u_n(x) with known Fourier transforms U_n(\xi), which as n \to \infty approximates the behaviour of a \delta functional.

    A suitable sequence can be constructed from Gaussians with decreasing spread parameters (who's FTs form a sequence of Gaussians with increasing spread parameters).

    Then to prove (1) you consider:

    \lim_{n \to \infty} \int f(x)u_n(x) e^{i 2 \pi \xi x}dx \  = \lim_{n \to \infty }(F*U_n)(\xi)

    By construction the limit on the left hand side is f(0), and on the right hand side you can exchange the limit and integral and so find it goes to right hand side of (1).

    (Of course this is much more direct it you run amok with generalised functions or distributions, when you use \delta(x) instead of \{ u_n(x)\}, and 1 instead of \{U_n(\xi)\})

    (And also this is modulo the odd factor depending on how the FT has been defined)

    RonL
    Last edited by CaptainBlack; April 17th 2008 at 01:18 PM. Reason: did not understand my earlier argument
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