we can represent any analytic function with convolutional graph filters

Data

Fact

Suppose $f$ is an analytic function. Then there exist coefficients $h_{i}$ such that

\begin{aligned} h (λ) & = \sum_{k = 0}^{\infty} h_{k} Λ^{k} \\ = f (0) + λ f^{'} (0) + \frac{λ^{2}}{2!} f^{″} (0) + \frac{λ^{3}}{3!} f^{‴} (0) + \dots \\ = f \end{aligned}

Proof

This follows immediately from the definition of an analytic function

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