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Verifying the Fourier Series is in Hilbert Space

  1. Nov 27, 2015 #1
    The text does it thusly:


    imgur link: http://i.imgur.com/Xj2z1Cr.jpg

    But, before I got to here, I attempted it in a different way and want to know if it is still valid.

    Check that [itex]f^{*}f[/itex] is finite, by checking that it converges.

    [tex]f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x + b_2^2sin^22x + ...[/tex]

    should converge provided the a's and b's are both less than 1 in absolute value.

    Is this also a valid way to show that our Fourier Series is an okay vector to work with?
  2. jcsd
  3. Nov 27, 2015 #2


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    What is it you want to show ? Orthogonality is OK, and normality is OK (provided you use the correct normalization constants). Completeness ? For all functions you can write the recipe to get the fourier transform.

    So yes, the Fourier basis is OK.
  4. Nov 29, 2015 #3


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