For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator $d/dx.$
I feel confunsed about the connection between the differential operator and spectral expansion of an arbitrary function. (In the example above, the connection between $d/dx$ and the Fourier expansion.) Why do people often say a differential operator (e.g., Laplacian) has a spectral expansion?
Thank you!