Dispersion of the Fibonacci and the Frolov point sets
For researchers in numerical integration and approximation theory, this provides optimal theoretical guarantees for point sets used in high-dimensional problems.
The paper proves that Fibonacci and Frolov point sets achieve optimal dispersion decay rate, enabling universal discretization of uniform norm for multivariate trigonometric polynomials. It also shows Fibonacci sets universally discretize all integral norms.
It is proved that the Fibonacci and the Frolov point sets, which are known to be very good for numerical integration, have optimal rate of decay of dispersion with respect to the cardinality of sets. This implies that the Fibonacci and the Frolov point sets provide universal discretization of the uniform norm for natural collections of subspaces of the multivariate trigonometric polynomials. It is shown how the optimal upper bounds for dispersion can be derived from the upper bounds for a new characteristic -- the smooth fixed volume discrepancy. It is proved that the Fibonacci point sets provide the universal discretization of all integral norms.