#### Conference abstracts

Session A1 - Approximation Theory

July 12, 17:00 ~ 17:35 - Room B3

## On the entropy numbers of the mixed smoothness function classes

### Vladimir Temlyakov

### University of South Carolina and Steklov Institute of Mathematics, United States - temlyakovv@gmail.com

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 talk is to present 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.