I originally asked this in the Calculus & Analysis forum. But perhaps this is better suited as a question in Abstract algebra.(adsbygoogle = window.adsbygoogle || []).push({});

For the set of all Dirac delta functions that have a difference for an argument, we have the property that:

[tex]\int_{ - \infty }^\infty {{\rm{\delta (x - }}{{\rm{x}}_1}){\rm{\delta (}}{{\rm{x}}_1}{\rm{ - }}{{\rm{x}}_0})d{x_1}} = {\rm{\delta (x - }}{{\rm{x}}_0})[/tex]

But the above equation can be seen as a convoltuion integral which can be solved with the Fourier transforms method. All three delta functions do exist in this set. And I'm told that the Fourier transform maps this set of Dirac deltas into this same set of Dirac deltas that have a difference as input. So does that gaurantee the existence of the Fourier transform for this set of Dirac deltas? I'm trying to determine if the FT for this set of Dirac delta functions is a necessity or optional added feature that could be used. And in my reading I'm told that the Fourier transform is an automorphism which transforms the deltas into themselves. Does that gaurantee the existence of the FT? What if you add the requirement that the convolution exist as seen in the above equation? If the convolutions exist and the FTs also exist in the same set of Dirac delta functions, does the above equation imply that the FT must also exist? Maybe this is the same as asking whether all automorphisms necessarily exist? Or maybe the FT is an equivalent automorphism to the convolution automorphism for this set of Dirac delta functions? Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Must the Fourier transform exist for Dirac delta functions?

**Physics Forums | Science Articles, Homework Help, Discussion**