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Fourier series and transforms

  1. Apr 19, 2007 #1
    Fourier series is a way to express a periodic function as a sum of complex exponentials or sines and cosines.. Is there actually a proof for the fact tat a periodic function can be split up into sines and cosines or complex exponentials?
  2. jcsd
  3. Apr 19, 2007 #2


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    It is fairly easy to show that any integrable periodic function can be approximated arbitrarily well by a sum of sines and cosines. The Fourier series is the limit of those approximations as the "error" goes to 0- except on a set of measure 0 for dis-continuous functions. I might point out that the other way is what's hard. It can be shown that some perfectly valid Fourier series converge to non-(Riemann)-integrable functions. That was why Lebesque integration had to be invented.
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