[nestes]

I'm surprised you're one of the only commenters to bring this up. I have an electrical engineering background -- for analysis, lots of systems are assumed to be either linear or very weakly nonlinear, and a lot of our signals are roughly periodic. Fourier transforms are a no-brainer.

Convolution turns into multiplication, differentiation wrt time of the complex exponential turns into multiplication by j*omega. I don't know about you, but I'd rather do multiplication than convolution and time derivatives.

As a corollary, once you accept "we use the Fourier representation because it's convenient for a specific set of common scenarios", the use of any other mathematical transform shouldn't be too surprising (for other problems).