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Fourier Series

  1. Oct 21, 2004 #1

    Zurtex

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    Hi, could someone give me an explanation of Fourier Series' please or a link that would give someone who has no idea about them a working grasp of what they are.
     
  2. jcsd
  3. Oct 21, 2004 #2
  4. Oct 22, 2004 #3

    Zurtex

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  5. Oct 22, 2004 #4
    I'll try to "sum up" (in a manner of speaking) an explanation i found in a textbook:

    "For problems with the periodic boundary conditions on the interval [tex]-L \leq x \leq L[/tex], we asked whether the following infinite series (a Fourier series) makes sense:
    [tex]f(x) = a_0 + \sum_{n=1}^\infty a_ncos(\frac{n\pi x}{L}) + \sum_{n=1}^\infty b_nsin(\frac{n\pi x}{L})[/tex]
    Does the infinite series converge? Does it converge to f(x)? Is the resulting infinite series really a solution of the partial differential equation? Mathematicians tell us that none of these questions have simple answers..."

    the mathworld site says "A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines", which i think is a good start. it doesn't have really good pics though because they only show 1 period of each function. take a copy of the square-wave function & move it to the right 2 units, so it jumps from -1 to +1 @ x=2, and do that infinitely many times in each direction. if you take the partial sums of the Fourier series for each curve you get approximations, shown in different colours in the pictures. if you do the whole infinite sum you get the periodic function exactly (except for certain functions). now can you see that trig functions, being periodic, would come in handy for approximating periodic functions? the only things we have to figure out are the [tex]a_ns[/tex] & [tex]b_ns[/tex], & there are formulas for those. it turns out that a function is the sum of its Fourier series if it's piecewise smooth (ie has continuous derivatives) on the interval [-L,L]
     
  6. Oct 22, 2004 #5

    matt grime

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    "piecewise smooth (ie has continuous derivatives)"

    Is that really the definition of piecewise smooth? I'm not convinced, you know.

    Expanding in terms of other functions is something you're ok with, Zurtex from Maclaurin and Taylor Expansions where we expand f(x) in terms of polynomials in x, and simple ones at that. We do that because we can easily find the coeffecients by differentiating.

    Similarly one can expand functions in terms of many other smaller fucntions. And if a function is periodic on R, or just defined on some interval [-L,L] from which one can make a periodic function by repeating it (we do not need continuity, and definitely not differentiability to do this stuff), we can use cos and sin.

    Finding these coefficients is "easy" because of various results about integrals.

    It so happens that sins and cosines are dense in the space of functions hence it is a genuine expansion.

    One can do this with lots of other functions too, and if you do quantum mechanics later you'll see how (or Hilbert Spaces, in pure terms).
     
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