First of all, I don’t have a solid math background, so a less general, but easier answer is preferable.
In “The Emerging Field of Signal Processing on Graphs” by Shuman, et al., they describe how the Graph Fourier Transform is similar to the conventional Fourier transform (conventional in the sense that this is how Fourier transform is usually explained in coursework). From the paper:
What is the intuition behind using the Laplacian for the Fourier transform? I thought about the classical Fourier transform as a change of domain from time to frequence, but not in relation to the eigenfunctions of the Laplace Operator.
So far, I tried to reason about Graph Fourier transform by comparing to classical Fourier Transform, but maybe I should think about classical Fourier Transform in terms of its eigenfunctions of the Laplace operator?
Update: Here in “Graph Structured Data Viewed Through a Fourier Lens” by Venkatesan Ekambaram, I found this:
There has been quite some work in the literature pursuing an ideal Fourier-like basis for signals defined on graphs, focusing particularly on the properties of Laplacian matrix eigenvectors. Interestingly, these eigenvectors are analogous to sinusoids in the time domain in that they have a natural signal-frequency interpretation.
This is very related to my questions/issue. I am wondering whether I should interpret Fourier transform based on the eigenvectors of the Laplacian which so happen to be sinusoids in the time domain instead of thinking about Fourier being defined through sinusoids.