Fourier Trasform of Delta functions

[muzialis]

Hi All,

I am trying to understand more rigorously why the Fourier transform of a constant functions equals the Dirac delta distribution.

I found somewhere this is justified by imposing the self-adjointness, so that under a duality pairing <..,..> and indicating with F(f) the transform of a function f, it is required that

<F(δ), g> = < δ , g >

If g equals the constant unitary function my source, http://en.wikipedia.org/wiki/Dirac_delta_function#Fourier_transform, quotes,

<1, F (g)> = g (0)= < δ , g >

I understand the second equality, but not sure about the first...

Many thanks for your help

 

[Edgardo]

I found these links:
https://nrich.maths.org/discus/messages/114352/115577.html?1172408092
Tutorial on the Dirac delta-function and Fourier transformation (see Appendix C and solutions, Exercise C7)