integral Lipschitz filter

[[concept]]

integral lipschitz filter

Let $h(\lambda )$ be the spectral representation of a convolutional graph filter. $h(\lambda )$ is integral lipschitz on interval $T\subset \mathbb{R}$ means there exists some $c\in \mathbb{R}$ such that $|h(\lambda )-h({\lambda }^{\prime })|\le \frac{C|{\lambda }^{\prime }-\lambda |}{\frac{1}{2}|\lambda +{\lambda }^{\prime }|}\forall \lambda ,{\lambda }^{\prime }$ ie, $h(\lambda )$ is lipschitz with a constant inversely proportional to the interval’s midpoint

Letting ${\lambda }^{\prime }\to \lambda$, we get $\lambda h^{\prime }(\lambda )\le c⟹h^{\prime }(x)\le \frac{c}{\lambda }\to 0,\lambda \to \infty$ This means that the filter can’t change for large $\lambda$.

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