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Fourier Question?

  1. Sep 19, 2005 #1
    Actually this might not be a fourier question, but it certainly reminds of Fourier series.

    Suppose,

    [tex]
    \sum_{n=0}^\infty a_n \, g_n(x) = 0
    [/tex]

    Does it necessarily follow that [itex]a_n = 0 \: \forall n[/itex]? If so, please provide a proof. If not, a counterexample would be helpful. If not, can I deduce anything about the the coefficients?

    A similar formula,
    [tex]
    f(x) g(y) = 0
    [/tex]

    only implies that each function must be a constant.
    [tex]
    \sum_n f_n(x) \, g_n(y) = 0
    [/tex]
    Under a sum, my guess is that we can't say anything about each of the functions.
     
  2. jcsd
  3. Sep 19, 2005 #2

    Tide

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    The [itex]a_n[/itex] will necessarily be zero only if the basis functions [itex]g_n[/itex] are mutually orthogonal (linearly independent).
     
  4. Sep 19, 2005 #3
    Haha! I knew I had seen that sum before! Independence!

    lol...thanks.
     
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