Why do we use trig functions in Fourier transforms, and not other periodic functions?

Why, when we perform Fourier transforms/decompositions, do we use sine/cosine waves (or more generally complex exponentials) and not other periodic functions? I understand that they form a complete basis set of functions (although I don’t understand rigourously why), but surely other period functions do too?

Is it purely because sine/cosine/complex exponentials are convenient to deal with, or is there a deeper reason? I get the relationship these have to circles and progressing around them at a constant rate, and how that is nice and conceptually pleasing, but does it have deeper significance?

Answer

The Fourier basis functions eiωx are eigenfunctions of the shift operator Sh that maps a function f(x) to the function f(xh):
eiω(xh)=eiωheiωx
for all xR.

All of the incarnations of the Fourier transform (such as Fourier series and the discrete Fourier transform) can be understood as changing basis to a basis of eigenvectors for a shift operator.

It is possible to consider other operators, which have different eigenfunctions leading to different transforms. But this shift operator is so simple and fundamental that it’s not surprising the Fourier transform turns out to be particularly useful.

Attribution
Source : Link , Question Author : user6873235 , Answer Author : littleO

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