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Are fourier series unique?

  1. Nov 7, 2004 #1


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    For a given function with a certain finite period, is there only one set of fourier series coefficients [tex]a_n[/tex] and [tex]b_n[/tex]? The reason I ask is, I was doing a problem where it asked for the coefficients for a certain odd function, and then it asked for the coefficients for that same function shifted up by a constant. Are all the [tex]b_n[/tex]'s the same, and [tex]a_0[/tex] just twice the constant? I tried real quick using the definition to see if this came out the same, and it didn't, but I might have made a mistake. Could there be two series for the same function?
  2. jcsd
  3. Nov 8, 2004 #2

    matt grime

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    Given a function, the coefficients are unique, though the converse is false.

    Consider the example of f(x) and f(x)+k, for some k, which is how I read your query.

    Then, since the integral of ksin(nx) and kcos(mx) are zero over the interval, it follows teh non-constant terms are the same, and the constant term is then the integral of f(x)+k, which is the original constant integral plus k times the length of the interval. Note, I haven't allowed for dividing by 2pi or anything since that is a non-canonical choice, and I hope you can fill in the constants properly.
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