Stochastic Atomic Decompositions in a Wavelet Dictionary

We consider random functions defined by atomic decompositions in a wavelet dictionary. The scale and dilation of the wavelet atoms are not dyadic constraints, but the function is modelled as a sum of wavelet functions at arbitrary positions and scales. The locations of the wavelet atoms and the magnitudes of their coefficients are chosen with respect to a certain marked Poisson process model, allowing intuitive notions about the functions genuinely to be modelled. We explore the properties of realizations from the proposed models by investigating the relationships between the parameters of the models and the Besov spaces within which realizations fall. Such relations may be seen as giving insight into the meaning of the Besov space parameters themselves. Moreover, although we do not elaborate on this, the established relations make it possible in principle to incorporate prior knowledge about the function’s regularity into statistical applications of the proposed models.