Parseval's identity

The Parseval relation holds for the graph Fourier transform,^[5] that is, for any ${\displaystyle f,h\in \mathbb {R} ^{N}}$

${\displaystyle \langle f,h\rangle =\langle {\hat {f}},{\hat {h}}\rangle .}$

This gives us Parseval's identity:^[3]

${\displaystyle \sum _{i=1}^{N}|f(i)|^{2}=\|f\|_{2}^{2}=\langle f,f\rangle =\langle {\hat {f}},{\hat {f}}\rangle =\|{\hat {f}}\|_{2}^{2}=\sum _{l=0}^{N-1}\left|{\hat {f}}\left(\lambda _{l}\right)\right|^{2}.}$

Generalized convolution operator

The definition of convolution between two functions ${\displaystyle f}$ and ${\displaystyle g}$ cannot be directly applied to graph signals, because the signal translation is not defined in the context of graphs.^[4] However, by replacing the complex exponential shift in classical Fourier transform with the graph Laplacian eigenvectors, convolution of two graph signals can be defined as:^[3]

${\displaystyle (f*g)={\mathcal {I}}{\mathcal {G}}{\mathcal {F}}[{\mathcal {G}}{\mathcal {F}}[f]\cdot {\mathcal {G}}{\mathcal {F}}[g]],}$

${\displaystyle (f*g)(i)=\sum _{l=0}^{N-1}{\hat {f}}(\lambda _{l}){\hat {g}}(\lambda _{l})\mu _{l}(i).}$

Generalized translation operator

As previously stated, the classical translation operator ${\displaystyle T_{v}}$ cannot be generalized to the graph setting. One way to define a generalized translation operator is through generalized convolution with a delta function centered at vertex ${\displaystyle n}$:^[2]${\displaystyle \left(T_{n}f\right)(i)={\sqrt {N}}\left(f*\delta _{n}\right)(i){=}{\sqrt {N}}\sum _{l=0}^{N-1}{\hat {f}}\left(\lambda _{l}\right)u_{l}^{*}(n)u_{l}(i),}$

where ${\displaystyle \delta _{i}(n)={\begin{cases}1,&{\text{if }}i=n,\\0,&{\text{otherwise.}}\end{cases}}}$

The normalization constant ${\displaystyle {\sqrt {N}}}$ ensures that the translation operator preserves the signal mean,^[4] i.e.,

${\displaystyle \sum _{i=1}^{N}(T_{n}f)(i)=\sum _{i=1}^{N}f(i).}$

Image compression

Representing signals in frequency domain is a common approach to data compression. As graph signals can be sparse in their graph spectral domain, the graph Fourier transform can also be used for image compression.^[6][7]

Graph noise reduction

Similar to classical noise reduction of signals based on Fourier transform, graph filters based on the graph Fourier transform can be designed for graph signal denoising.^[8]

Data classification

As the graph Fourier transform enables the definition of convolution on graphs, it makes possible to adapt the conventional convolutional neural networks (CNN) to work on graphs. Graph structured semi-supervised learning algorithms such as graph convolutional network (GCN), are able to propagate the labels of a graph signal throughout the graph with a small subset of labelled nodes, theoretically operating as a first order approximation of spectral graph convolutions without computing the graph Laplacian and its eigendecomposition.^[9]