Adaptive Martingale Approximations

H-systems are those orthonormal systems which allow computation of conditional expectations via a Fourier expansion. These systems provide natural approximations to continuous stochastic processes and we indicate how they could be used to perform lossy compression on a set of given signals. More specifically, we show how to construct, for a given random variable, an adaptive martingale approximation by means of generalized Haar functions. We also indicate how the construction can be extended to a given random vector. A generalized multiresolution analysis algorithm is also described and numerical examples are provided.