On localisation and uncertainty mesures on graphs

Due to the appearance of data on networks such as internet or Facebook, the number of applications of signal on weighted graph is increasing. Unfortunately, because of the irregular structure of this data, classical signal processing techniques are not applicable. In this paper, we examine the windowed graph Fourier transform (WGFT) and propose ambiguity functions Ag to analyze the spread of the window in the vertex-frequency plane. We then observe through examples that there is a trade-off between the vertex and frequency resolution. This matches our intuition form classical signal processing. Finally, we demonstrate an uncertainty principle for the spread of the ambiguity function: ‖Ag‖1 ‖Ag‖∞ > 1 μ2 . We verify with examples that this principle is sharp for the extreme values of μ and emphasize the difference between the generalized graph ambiguity function and the classical one. We finish with demonstration of some Young and Hausdorff-Young like inequalities for graphs.