we can write a graphon in the basis of its shift operator

[[concept]]

Theorem

We can write the graphon $W$ in the basis $\{{\phi }_i\}$ for its graphon shift operator as $W(u,v)={\sum }_{i=0}^1{\lambda }_i{\phi }_i(u){\phi }_i(v)$ (compare this to $S=V\Lambda V^{\ast }$ for a spectral graph filter)

Proof

Since a graphon shift operator is a self-adjoint Hilbert-Schmidt integral operator, this is a direct consequence of the spectral theorem for self-adjoint compact operators on Hilbert spaces.

see graphon shift operator eigenvalues

Review

dsg

We can write the graphon $W$ in the basis of its {graphon shift operator}

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