Connections between numerical integration, discrepancy, dispersion, and universal discretization
It provides a unified perspective for researchers working on approximation theory and numerical analysis, but is primarily a survey of existing results.
This survey connects results from numerical integration, discrepancy, dispersion, and universal discretization, highlighting recent findings such as the utility of fixed volume discrepancy for dispersion bounds and the effectiveness of low-dispersion point sets for universal discretization of trigonometric polynomials.
The main goal of this paper is to provide a brief survey of recent results which connect together results from different areas of research. It is well known that numerical integration of functions with mixed smoothness is closely related to the discrepancy theory. We discuss this connection in detail and provide a general view of this connection. It was established recently that the new concept of {\it fixed volume discrepancy} is very useful in proving the upper bounds for the dispersion. Also, it was understood recently that point sets with small dispersion are very good for the universal discretization of the uniform norm of trigonometric polynomials.