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).
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).