On the entropy numbers of the mixed smoothness function classes
This work addresses fundamental open problems in approximation theory, providing incremental advances in understanding the complexity of mixed smoothness function classes.
The paper tackles the problem of bounding entropy numbers for multivariate functions with mixed smoothness, developing a new method based on greedy approximation to prove upper bounds and using volume estimates for lower bounds, resulting in improved theoretical bounds for these function classes.
Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop a new method of proving the upper bounds for the entropy numbers. This method is based on recent developments of nonlinear approximation, in particular, on greedy approximation. This method consists of the following two steps strategy. At the first step we obtain bounds of the best m-term approximations with respect to a dictionary. At the second step we use general inequalities relating the entropy numbers to the best m-term approximations. For the lower bounds we use the volume estimates method, which is a well known powerful method for proving the lower bounds for the entropy numbers. It was used in a number of previous papers.