On lattice rules, approximation and Lebesgue constants

In this talk I start with introducing lattice rules for numerical integration with the trigonometric degree as quality criterion. I focus on Fibonacci lattice rules and lattice rules with minimal number of points for the trigonometric degree. Then I consider the points of these lattice rules for approximation and introduce the trigonometric approximation degree. This approximation degree is calculated for the lattices used throughout this talk. To investigate the quality of point sets for approximation the Lebesgue constant is often used. The Lebesgue constant for trigonometric approximation for 1- and 2-dimensional lattices is investigated. We reveal some nice structures and make the link between the Dirichlet kernel and the reproducing kernel that was used to obtain minimal lattice rules in two dimensions.status: publishe