A Characterization of Higher Order Nets Using Weyl Sums and Its Applications

Point sets referred to as (t, α, β, n, m, s)-nets were recently intro- duced and shown to generalize both digital (t, α, β, n × m, s)-nets and classical (t, m, s)-nets. Their definition captures the geometrical properties of their digital analogue, which has recently been shown to yield quadrature points for quasi- Monte Carlo rules which can achieve arbitrary high convergence rates of the integration error for sufficiently smooth functions. In this paper, we character- ize (t, α, β, n, m, s)-nets using Weyl sums generalizing the analogous result for (t, m, s)-nets. As an application of this characterization we study numerical integration using such higher order nets. It is shown that for functions having square integrable mixed partial derivatives of order α in each variable, integration errors converge at a rate of N−(α−1)+δ for any δ > 0, establishing that (t, α, β, n, m, s)-nets can exploit the smoothness of the function under consideration. The characterization is consequently employed to study the randomization of (t, α, β, n, m, s)-nets and the application of randomized (t, α, β, n, m, s)-nets to numerical integration. It is found that the root mean-square error converges at a rate of N−(α− 2 1 )+δ for any δ > 0, improving on the result on integration errors associated with (t, α, β, n, m, s)-nets. As a further application, it can be used for the construction of new (t, α, β, n, m, s)-nets itself: We introduce an analogue of the (u, u + v)-construction for digital (t, α, β, n × m, s)-nets and (t, m, s)-nets.