Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Are fourier series unique?

  1. Nov 7, 2004 #1


    User Avatar
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Are fourier series unique?
  1. Fourier Series (Replies: 1)

  2. Fourier Series (Replies: 1)

  3. Fourier series (Replies: 1)

  4. Fourier Series (Replies: 1)