graphon fourier transform

[[concept]]

Graphon Fourier transform

The graphon fourier transform of a graphon signal $(W,\mathcal{X})$ is a functional $\hat{\mathcal{X}}=WFT(\mathcal{X})$ defined as ${\hat{\mathcal{X}}}_j=\hat{\mathcal{X}}({\lambda }_j)={\int }_0^1\mathcal{X}(u){\phi }_j(u)du$ where ${\lambda }_j$ are the eigenvalues of $W$ and $\{{\phi }_i\}$ are the eigenfunctions.

Note

Since the ${\lambda }_j$ are countable, the WFT is always defined. (see spectral theorem for self-adjoint compact operators on Hilbert spaces)

see also inverse graphon fourier transform

Review

dsg

Why is the graphon fourier transform always defined? -?- The eigenvalues ${\lambda }_j$ are countable

Why are the eigenvalues of a graphon signal countable? -?- This is a direct application of the spectral theorem for self-adjoint compact operators on Hilbert spaces

References

Mentions

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