A characterization of higher order nets using Weyl sums and its applications

Point sets referred to as $(t,\alpha,\beta,n,m,s)$-nets were recently introducedand shown to generalize both digital $(t,\alpha,\beta, n \times m,s)$-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 characterize $(t,\alpha,\beta,n,m,s)$-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 $\alpha$ in each variable, integration errors converge at a rate of $N^{-(\alpha-1)+ \delta}$ for any $\delta > 0$, establishing that $(t,\alpha,\beta, n, m, s)$-nets can exploit the smoothness of the function under consideration. The characterization is consequently employed to study the randomization of $(t,\alpha,\beta,n,m,s)$-nets and the application of randomized $(t,\alpha,\beta,n,m,s)$-nets to numerical integration. It is found that the root mean-square error converges at a rate of $N^{-(\alpha - \frac{1}{2}) + \delta }$ for any $\delta > 0$, improving on the result on integration errors associated with $(t,\alpha, \beta,n,m,s)$-nets. As a further application, it can be used for the construction of new $(t,\alpha, \beta,n,m,s)$-nets itself: We introduce an analogue of the $(u,u + v)$-construction for digital $(t,\alpha, \beta, n \times m,s)$-nets and $(t,m,s)$-nets.