On the robustness of certain norms

We study a family of norms defined for functions on an interval. These norms are obtained by taking the $p$-norm of the Volterra operator applied to the function. The corresponding distances have been previously studied in the context of comparing probability measures, and special cases include the Earth Mover's Distance and Kolmogorov Metric. We study their properties for general signals, and show that they are robust to additive noise. We also show that the norm-induced distance between a function and its perturbation is bounded by the size of the perturbation, and that the distance between one-dimensional projections of a two-dimensional function is bounded by the size of the difference in projection directions. The results are illustrated in numerical experiments.