Strong Tractability of Quasi-Monte Carlo Quadrature Using Nets for Certain Banach Spaces

We consider multivariate integration in the weighted spaces of functions with mixed first derivatives bounded in $L_p$ norms and the weighted coefficients introduced via $\ell_q$ norms, where $p,q\in[1,\infty]$. The integration domain may be bounded or unbounded. The worst‐case error and randomized error are investigated for quasi‐Monte Carlo quadrature rules. For the worst‐case setting the quadrature rule uses deterministic $((T_u),s)$‐sequences in base b, and for the randomized setting the quadrature rule uses randomly scrambled digital $((T_u),m,s)$‐nets in base b. Sufficient conditions are found under which multivariate integration is strongly tractable in the worst‐case and randomized settings, respectively. Similar results hold for the Banach spaces of finite‐order weights. Results presented in this article extend and improve upon those found previously.