I am currently learning about the Fourier Transform and the associated Fourier Analysis.
So far I realize that it has a number of applications, but more than that it seems to be central to Functional Analysis.
I am trying to understand why this is the case.
In particular, I was wondering whether it is the Fourier Transform itself that is so fundamental, or whether it is only the process of a linear transform that makes it seem "special". So for example, the Laplace Transform is also linear, but my impression is that it has a less prominent role in functional analysis.
Edit: I just found this question was also posed on Math Overflow some time ago, and received a couple of highly interesting answers (for those that are interested, here is the link).