Sampling discretization error of integral norms for function classes with small smoothness

Abstract

We consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on the integral norm discretization, we consider the absolute error setting. We demonstrate how known results from two areas of research -- supervised learning theory and numerical integration -- can be used in sampling discretization of the square norm on different function classes. We prove a general result, which shows that the sequence of entropy numbers of a function class in the uniform norm dominates, in a certain sense, the sequence of errors of sampling discretization of the square norm of this class. Then we use this result for establishing new error bounds for sampling discretization of the square norm on classes of multivariate functions with mixed smoothness.