A measure theoretic approach to Lipschitz regularity and its Haar type wavelet analysis

The $\alpha$−Lipschitz character of a time series or an image summarizes, in the single parameter α, some persistence properties of the original function modeling the given signal. Such is the case of the Hurst exponent widely used in hydrology and other sciences. It is known that the continuous wavelet transform is useful to characterize the $\alpha$−Lipschitz condition of a function. Recent results show that, in one dimension, the behavior of the Haar coefficients characterize the $\alpha$−Lipschitz regularity of functions with respect to the induced dyadic metric. In this note, we extend the characterization of dyadic Lipschitz regularity of functions to non-atomic probability spaces, using generalized Haar systems. We also provide some two dimensional examples that can be designed and used to reflect specific textures in images.