Windowed Fractional Fourier Transform on Graphs: Fractional Translation Operator and Hausdorff-Young Inequality

Designing transform method to identify and exploit structure in signals on weighted graphs is one of the key challenges in the area of signal processing on graphs. So we need to account for the intrinsic geometric structure of the underlying graph data domain. In this paper we generalize the windowed fractional Fourier transform to the graph setting. First we review the windowed fractional Fourier transform and introduce spectral graph theory. Then we define a fractional translation operator with interesting property for signals on graphs. Moreover, we use the operator to define a windowed graph fractional Fourier transform, and explore the reconstruction formula. Finally, the Hausdorff-Young inequality established on this new transform is obtained.