The Multi-resolution Norm for Non-parametric Regression and Inverse Problems

Abstract

We study the application of the multi-resolution norm, which measures the maximal size of scaled local means of a function, in the problems of non-parametric regression and deconvolution. Our main result is the derivation of convergence rates for Tikhonov regularization, where we use the multi-resolution norm as the similarity term and a homogeneous Sobolev norm for the regularization term. The results are based on an asymptotic estimate for the norm of samples of a Gaussian random variable as the sample size tends to infinity, and an interpolation inequality for the multi-resolution norm and Sobolev norms. This is a joint work with Housen Li and Axel Munk (University of Göttingen).