Tensor sparsity of solutions of high dimensional PDEs

We introduce a class of Bernstein-Durrmeyer operators with respect to an arbitrary measure on a multi-dimensional simplex. These operators generalize the well-known Bernstein-Durrmeyer operators with Jacobi weights. A motivation for this generalization comes from learning theory. In the talk, we discuss the question which properties of the measure are important for convergence of the operators. We show that uniform convergence holds for all continuous functions on the simplex if and only if the underlying measure is strictly positive on the simplex. We describe a more special class of strictly positive measures for that we can give an estimate for the rate of convergence. Finally, we discuss convergence on the support of non-strictly positive measures.