We can write the graphon $W$ in the basis $\{{\phi }_i\}$ for its graphon shift operator as

(compare this to $S=V\mathrm{\Lambda }{V}^{\ast }$ for a spectral graph filter)

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.