the spectral graph filter operates on a signal pointwise

The spectral representation of a convolutional graph filter $\widehat{H(S)}$ is pointwise in the spectral domain. ie ${\hat{y}}_i=\sum _{k=0}^{K-1}{h}_k{\lambda }_i^k{\hat{x}}_i$. This follows immediately from the fact that $\mathrm{\Lambda }$ is diagonal.