Error estimates for multivariate regression on discretized function spaces

In this paper, we will discuss the discretization error for the regression setting and derive error bounds relying on the approximation properties of the discretized space. Furthermore, we will point out how the sampling error and the discretization error interact and how they can be balanced appropriately. We will present two examples based on tensor product spaces (sparse grids, hyperbolic crosses) which provide a suitable approach in the case of large sample sets in moderate dimensions.