Sampling inequalities for sparse grids

Abstract

Sampling inequalities play an important role in deriving error estimates for various reconstruction processes. They provide quantitative estimates on a Sobolev norm of a function, defined on a bounded domain, in terms of a discrete norm of the function's sampled values and a smoothness term which vanishes if the sampling points become dense. The density measure, which is typically used to express these estimates, is the mesh norm or Hausdorff distance of the discrete points to the bounded domain. Such a density measure intrinsically suffers from the curse of dimension. The curse of dimension can be circumvented, at least to a certain extend, by considering additional structures. Here, we will focus on bounded mixed regularity. In this situation sparse grid constructions have been proven to overcome the curse of dimension to a certain extend. In this paper, we will concentrate on a special construction for such sparse grids, namely Smolyak's method and provide sampling inequalities for mixed regularity functions on such sparse grids in terms of the number of points in the sparse grid. Finally, we will give some applications of these sampling inequalities.