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I Generalized version of the Fourier Transform

  1. Aug 31, 2016 #1
    Hello everyone,

    I was trying to develop a sort of generalized version of the Fourier Transform. My question in particular is:
    Given a function [itex]f(x,u)[/itex], is there a function [itex]g(x,u)[/itex] with [tex]\int_{-\infty}^\infty f(x,u)g(x,u')\mathrm{d}x=\delta(u-u')[/tex]

    For [itex]f(x,u)=e^{2\pi ixu}[/itex] the solution would be [itex]g(x,u)=\frac{1}{2\pi}e^{-2\pi ixu}[/itex]. Are there other pairs with this property?

    Thank you for your help :)
  2. jcsd
  3. Aug 31, 2016 #2

    Dr Transport

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    look at the theory of Sturm-Liouville functions
  4. Aug 31, 2016 #3


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