graphon fourier transform

[[concept]]

Graphon Fourier transform

The graphon fourier transform of a graphon signal $(W, X)$ is a functional $\hat{X} = W F T (X)$ defined as

{\hat{X}}_{j} = \hat{X} (λ_{j}) = \int_{0}^{1} X (u) φ_{j} (u) d u

where $λ_{j}$ are the eigenvalues of $W$ and ${φ_{i}}$ are the eigenfunctions.

Note

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

Review

#flashcards/math/dsg

Why is the graphon fourier transform always defined?
-?-
The eigenvalues $λ_{j}$ 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

File
2025-03-31 lecture 16
2025-04-02 lecture 17
2025-04-07 lecture 18
bandlimited convergent graph signals converge in the fourier domain
bandlimited graphon signal
inverse graphon fourier transform
spectral representation of graphon convolutions

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