Sampling and reconstruction of convex functions

Abstract

We discuss optimal recovery for classes of multivariate convex functions from given point samples, as well as the sampling numbers of these classes, corresponding to optimal sample choices. Upper and lower bounds for either variant are established when the reconstruction error is measured in $L_p$ for $1\leq p\leq \infty$. These bounds match, sometimes up to logarithmic factors, and therefore characterize the respective optimal rate of decay. For classical smoothness classes such as Sobolev, H\"older or Besov spaces, it is well known that the optimal decay rate of sampling numbers can be achieved by sampling on uniform tensor product grids and using linear methods of reconstruction, such as piecewise polynomial interpolation. One of the main findings in this paper is that for classes of convex functions, these procedures generally produce suboptimal rates, except when $p=1$ and $p=\infty$, and are outperformed by nonlinear reconstruction methods that do not employ tensor product grids.