Generalized version of the Fourier Transform

[klpskp]

Hello everyone,

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

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

Thank you for your help :)

 

[Dr Transport]

look at the theory of Sturm-Liouville functions

 

[Svein]

What you are looking for is the concept of an orthonormal basis in a Hilbert space (https://en.wikipedia.org/wiki/Hilbert_space).