The Volume Estimates and Their Applications

Abstract : We prove new estimates for the entropy numbers of classes of multivariate functions with bounded mixed derivative. It is known that the investigation of these classes requires development of new techniques comparing to the univariate classes. In this paper we continue to develop the technique based on estimates of volumes of sets of the Fourier coefficients of trigonometric polynomials with frequencies in special regions. We obtain new volume estimates and use them to get right orders of decay of the entropy numbers of classes of functions of two variables with a mixed derivative bounded in the L1-norm. This is the first such result for these classes. This result essentially completes the investigation of orders of decay of the entropy numbers of classes of functions of two variables with bounded mixed derivative. The case of similar classes of functions of more than two variables is still open.