[zadwang]

Fourier basis is unique in that the complex exponential basis functions are the eigen vectors of the linear time invariant (LTI) systems. No other transform has this property. Many real world systems (circuits, communication channels, antennas, etc) are LTI. This property make sure for example, signals transmitted over different frequencies do not interfere. That is why Fourier transform is so useful and used instead of other transforms. There is also the connection with quantum physics, in using Fourier pair as wave functions of position and momentum, which other transforms don’t have.

[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).

[krackers]

>Fourier basis

Technically it's a specialized case of the laplace basis, right? I was always surprised that lots of courses jump directly from the (bilateral) fourier transform to the unilateral laplace transform without proper analysis of the most general case that is the bilateral laplace transform: https://en.wikipedia.org/wiki/Two-sided_Laplace_transform

[nestes]

That's true, Laplace corresponds to a basis of complex exponentials that can grow or decay in time instead purely imaginary exponentials. We restrict the Ae^[(a+jb)t] domain just to Ae^(jbt) for Fourier.

From an circuit analysis standpoint (your problem may be different), but exponentials that decay over time ("a" is negative) corresponds to loss in a circuit, whereas exponentials that grow over time ("a" positive) correspond to something blowing up (this is really a nonphysical result but generally means a circuit is going to oscillate on its own, without a source driving that response). I mostly do electromagnetics/passive RF types of problems, in which you generally want everything to be low-loss. In that case Fourier is perfect, especially since I typically care most about steady-state behavior.