fixed coefficients yield the same spectral response for both graphon and graph convolutions

[[concept]]

Important

Given the same (fixed) coefficients $h_{k}$ , define the graph convolution $H (S) x = \sum_{k = 0}^{K - 1} h_{k} S^{k} x$ . The spectral representation of that convolution is $h (λ) = \sum_{k = 0}^{K - 1} h_{k} λ^{k}$

this is the same as the spectral representation of graphon convolutions/function for the same coefficients.
The only difference is where the function is evaluated.
- For a graphon, we evaluate it at the graphon shift operator eigenvalues $λ_{j} (T_{W})$ for graphon signal $T_{H_{j}}$
- For a graph, this is the graph shift operator eigenvalues $λ_{j} (S)$ for the graph signal $H (S)_{j}$

Review

#flashcards/math/dsg

If we fix the coefficients $h_{k}$ for a graph convolution, the {ass||spectral representation||attribute} of the resulting graph filter {has||is the same||relationship} as the {ass||spectral representation||attribute} of {hha||the graphon convolution/filter||relation} with the same coefficients

Mentions

File	Last Modified
graph convolutions of bandlimited signals converge to the graphon convolution	2025-10-02
spectral representation of graphon convolutions	2025-10-02
the graph convolution converges to the graphon convolution in the spectral domain for a convergent sequence of graph signals	2025-10-02

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