I understand that Fourier analysis works (up to constant multiples) by considering the inner-product space $E$ of smooth functions $[-\pi,\pi] \to \mathbb C$ with inner product. . .
$\displaystyle (f,g) = \frac{1}{2 \pi}\int_{\pi}^\pi g(x) \overline{f}(x) dx$
. . . and then considering the symmetric operator $\Delta \colon E \to E$ called the Laplacian that works like. . .
$\displaystyle \Delta (f) = \frac{\partial ^2 f}{ \partial x^2}$
Then by the spectral theory every element of $E$ decomposes as an independent sum of eigenvectors of $\Delta$. But we can compute the eigenvectors to be the functions $e^{2\pi i n}$ for each $n \in \mathbb Z$. This gives the usual Fourier coefficients.
I am wondering if there is anything special about the choice of the Laplacian, other than its eigenvectors being immediately familiar. If we choose some other symmetric operator will we get a whole other version of the Fourier series? Could this be used to prove variants of the Plancheral formula for $\pi$ et cetera?
To narrow down the question, do Fourier series for different operators appear in any other area of mathematics under a different name?