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?

The Fourier basis functions $e^{i \omega x}$ are eigenfunctions of the shift operator $S_h$ that maps a function $f(x)$ to the function $f(x - h)$: $$ e^{i \omega (x-h)} = e^{-i\omega h} e^{i \omega x} $$ for all $x \in \mathbb R$.

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.

There is no direct mathematical reason to use sine/cosine/exponential. In fact you can define a similar decomposition using any orthogonal basis of the square integrable functions. For example you could decompose a function on an interval using the Legendre polynomials or in a more general sense take any sufficiently nice basis and do what is called a Wavelet transform. Most of the properties of the Fourier transform such as for example isometry will still hold, with proofs that are very much identical.

There are however a lot of indirect reasons to use sine/cosine/exponential, namely that they have a lot of nice and useful properties, mostly related to differentiation. Just to name a few of the top of my head:

• They are eigenfunctions of the differential operator. That is, they tend to reproduce under differentiation. $\frac{d}{dt} e^{ikt}$ again is a multiple of $e^{ikt}$. We can use this to solve differential equations by turning them into simple linear equations by taking the transform.
• They are the solutions of the simple harmonic oscillator $\ddot{f} = -kf$. This equation (or variations of it) turns up extremely often in physics and thus it is no wonder that Fourier series or transform is useful when dealing with such problems. (And indeed for other equations you will need a different transform)
• They are analytic and periodic. I know that you can turn any function on an interval into a periodic one, however since sine/cosine/exponential correspond to their own power series, they are kind of "naturally periodic".

There is some deeper significance from the point of view of representation theory.

For the Fourier transform on the circle, functions of the form $e^{ikx}$ (depending on the period/normalization) are precisely the characters, irreducible complex representations of the group $\mathbb T$ (which you can think of as ${\mathbf R}/{\mathbf Z}$, ${\mathbf R}/{\mathbf 2\pi \mathbf Z}$, or as the complex numbers of norm $1$, or another renormalization you prefer).

Functions $\sin(kx)$ and $\cos(kx)$ are the matrix coefficients of the irreducible real representations of the group.

Similarly, for the real line, $e^{i\xi x}$ are the irreducible complex unitary representations of the group $(\mathbf R,+)$, while $\sin(\xi x)$, $\cos(\xi x)$ are the matrix coefficients of the irreducible orthogonal real representations.

Representation theory gives a precise sense to the Fourier transform for any locally compact group (and probably more, but I'm no specialist), and in the abelian case, we have Pontryagin duality, which is responsible for the inverse Fourier transform.