Fractional Spectral Graph Wavelets and Their Applications

Table 1

The definitions of various transforms for classical signal processing and for graph signal processing.

Classical signal processing
Fourier transform	Fractional Fourier transform		Wavelet transform		Fractional wavelet transform
Discrete form of Fourier matrix: where diag(.) denotes a diagonal matrix formed from its vector argument.	where is the kth-order normalized Hermite function. Discrete form of fractional Fourier matrix:		with		with And è is the fractional order.
Graph signal processing
Algebraic signal processing (ASP) [40, 41]		Spectral graph theory [53, 54]
Graph Fourier transform	Graph fractional Fourier transform	Spectral graph Fourier transform	Spectral graph fractional Fourier transform	Spectral graph wavelet transform	Spectral graph fractional wavelet transform
where is obtained by the Jordan decomposition of the adjacency matrix W as follows: .	where where è is the fractional order.	where is obtained by the eigenvalue decomposition of the graph Laplacian matrix L as follows: , where D is the diagonal degree matrix and W is the adjacency matrix.	where where è is the fractional order.	where and L is the graph Laplacian matrix.	where and Lè is the graph fractional Laplacian matrix