Connections between numerical integration, discrepancy, dispersion, and universal discretization

Abstract

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 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.

On the fixed volume discrepancy of the Fibonacci sets in the integral norms

“…This paper is devoted to the study of a discrepancy-type characteristic -the fixed volume discrepancy -of a point set in the unit square Ω 2 := [0, 1) 2 . We refer the reader to the following books and survey papers on discrepancy theory and numerical integration [2], [7], [8], [19], [3] [5], [15], and [20]. Recently, an important new observation was made in [16].…”