Hyper-Differential Operator-Based Graph Fractional Fourier Transform: Novel Approaches and Applications

This paper introduces novel approaches and applications for hyper-differential operator-based graph fractional Fourier transform (HGFrFT), which provides a more flexible and efficient framework for graph signal processing. Its mathematical foundation is derived through the relationship between the graph Fourier transform and the fractional graph Fourier transform, thereby enhancing its ability to represent the underlying structure of graph signals. Specifically, four solutions to the Sylvester equation are discussed, and two fixed hyperdifferential operator matrices are introduced to define the HGFrFT. Furthermore, the applications of the HGFrFT in vertex-frequency analysis and optimal filter design are explored. Finally, the effectiveness of the HGFrFT is validated in electrocardiogram signal classification, demonstrating its applicability to real-world signal processing tasks. The theoretical framework and experimental results collectively advance fractional operator theory in graph domains.