Multiscale regularization in Besov spaces

There has been great research interest in thresholding methods for nonlinear wavelet regression over spaces of smooth functions. Near-minimax convergence rates were in particular established for simple hard and soft thresholding rules over Besov and Triebel bodies. We propose an alternative approach where nonstandard thresholding rules in dual spaces are obtained in possibly non-Gaussian noise situations, using a functional regularization framework. This method provides statistical prior models associated with the considered functional spaces. Connections with nonsmooth regularization using exponential power prior distributions are finally presented.